https://www.youtube.com/watch?v=14SaROmQKCI&feature=emb_logo

How these systems can transit between different order states.

00:05 actually the first time i managed to set
00:07 up
00:07 everything in time but i started a half
00:11 half an hour earlier to set up the
00:13 webcam and everything
00:15 so we have a professional equipment here
00:16 so that’s really a video conference
00:19 conferencing system and uh
00:22 but it has like 100 different cables
00:26 and a couple of devices only to be
00:28 connected in the right way
00:29 to make things work and but today i
00:33 managed
00:33 managed to set it up in time
00:36 okay so then let’s start our lecture
00:40 today
00:41 and um so
00:44 before i go on i’d like to give you a
00:47 reminder
00:48 of actually uh what we did last time
00:52 uh because that’s connected to what
00:54 we’re doing today
00:56 let me just share the screen
01:08 so there we go ah

slide 1

01:12 works flawlessly perfect okay great
01:16 uh so just a little reminder of last uh
01:18 kind lecture
01:19 the last time we started thinking about
01:22 how order
01:23 emerges a non-equilibrium system
01:27 and the key insight from last in the
01:30 last lecture is that it’s
01:32 many situation boils down to a balance
01:34 between
01:35 noise that create disorder
01:39 and the propagation of information about
01:43 interactions through the system
01:47 so in equilibrium conditions and the
01:49 thermal equilibrium this is formalized
01:52 by the entropy and the energy uh that
01:55 then give rise to the free energy that
01:57 we need to minimize
01:58 and then we minimize the free energy we
02:01 kind of
02:01 know what is happening yeah and when we
02:05 look at this equilibrium system mass
02:08 times are these pointers
02:09 also called xy model we saw that
02:13 actually in all dimensions smaller or
02:16 equal to two
02:18 fluctuations if we prepare the system in
02:21 a homogeneous
02:22 state where all pointers point of all
02:25 arrows point in the same direction
02:28 if we twist one arrow or if we perturb
02:31 the system
02:32 yeah then this destabilizes
02:36 the order on the large
02:39 scale yeah so we calculated how such
02:42 perturbations propagate through the
02:45 system that we saw that they build up
02:47 the the longer you go through the system
02:50 therefore this
02:51 distance r to infinity that we had last
02:53 time
02:54 this has formalized this idea that you
02:56 cannot build up a long range
02:58 order in uh equilibrium by the mermain
03:01 wagner theorem that basically says that
03:04 even if you had long range order it
03:06 doesn’t cost
03:07 any energy to have a very slow
03:10 perturbation
03:11 of your spins and this very slow
03:14 perturbation
03:15 is called the gold stone mode you can
03:17 always get these gold stone modes
03:20 in equilibrium conditions if your
03:21 symmetry so if your
03:23 spin or your your microscopic degrees of
03:26 freedom
03:27 are continuous now then we went on to
03:31 equilibrium
03:32 conditions uh non-equilibrium systems
03:34 and we thought
03:35 you know so about what is now if these
03:38 arrows are moving
03:39 now they’re not only pointing in a
03:41 direction but they’re also moving in the
03:43 same direction
03:44 then information about
03:48 the alignment of uh
03:51 or the direction does not only spread
03:54 diffusively so very inefficiently
03:57 equilibrium system but it can spread
04:01 very quickly through flows convective
04:04 flows
04:05 through the entire system and thereby
04:08 give rise to long-range order
04:10 even four dimensions
04:13 smaller or equal to two now so
04:16 in non-equilibrium systems we have
04:18 possibilities to spread the information
04:21 about alignment or to suppress
04:24 fluctuation to suppress
04:25 noise that we don’t have in equilibrium
04:28 conditions
04:29 and therefore we can get ordered states
04:32 and non-equilibrium systems
04:33 even if we cannot have them in very
04:36 similar
04:36 equilibrium systems

slide 2

04:40 so today i want to go
04:43 one step further further one step
04:46 further and one step back
04:47 actually um so before
04:50 let’s let’s take a step back before i
04:52 begin with today’s lecture
04:54 uh just to see where we are in the
04:56 lecture so we just
04:57 now got the idea of how order
05:01 arises in non-equilibrium systems today
05:04 we’ll
05:04 talk how these systems can

05:07 transit between different order states
05:10 so we’ll be talking about bifurcations
05:13 and phase transitions
05:15 and this
05:18 lecture today in this lecture today we
05:20 will focus on
05:21 very large systems where this noise is
05:24 not
05:25 this noise term that we have it’s psi is
05:28 not
05:28 important and in the next lectures
05:31 two or three or two lectures or so we
05:34 will
05:35 take into account what does noise do to
05:38 order states to do the transitions
05:40 between
05:40 faith and for this we will need methods
05:43 from renormalization group theory
05:45 that we will introduce here and then
05:49 we’re done with most of the
05:52 theoretical physics aspects of this
05:54 lecture after that we’ll
05:55 start taking a different approach and
05:58 ask how can we actually
05:59 see order in
06:03 big data sets somebody gives you a
06:05 terabyte of data how can you actually
06:07 identify these degrees of freedom
06:11 now if you have not only like a spin
06:13 system uh
06:14 one degree of freedom for each spin but
06:17 if you have 20
06:18 000 degrees of freedom how can you
06:19 actually see whether you have some kind
06:22 of collective
06:23 state in your system so that will so
06:26 we’ll have two or three
06:27 lectures on what is actually data
06:29 science
06:30 and then at the end of the lecture at
06:32 the end of january we’ll finish up by
06:34 putting it all together and see how we
06:36 can
06:36 switch between theory and data science
06:40 and back and how to actually generate
06:42 hypothesis
06:43 from the data sets that are currently
06:45 out there
06:47 okay so for the today’s lecture i
06:50 uh want to now not ask can we have
06:54 order or can we not have order i want to
06:57 ask what kind of order do we have
07:00 and uh to this i
07:03 uh want to come back to this formalism
07:07 that allows us to characterize generally
07:11 larger systems that have spatial degrees
07:14 of freedom
07:15 now that are spatial results so we have
07:17 a field
07:19 or an order parameter it’s called an
07:20 order parameter field phi
07:22 of x t now that gives us
07:26 uh so for each coordinate in space
07:29 this phi of x and t gives us
07:33 the value for example of some
07:35 concentration
07:37 or for example the number of infected
07:39 people
07:40 in dresden also yeah and then we
07:44 came up with something that is actually
07:46 standard in the literature
07:48 but it’s just a very inconvenient way of
07:50 actually writing down
07:52 partial differential equations is uh
07:55 that we said okay we can get
07:58 basically a large class of systems of
08:00 dynamics
08:01 if we take some functional f that
08:04 depends on phi
08:06 and we take the derivative with respect
08:09 to y
08:10 now with this we can create here on the
08:12 left hand side
08:13 all kinds of terms that are functions of
08:17 fine
08:17 and of derivatives of sine
08:20 and then we have our noise terms
08:25 here and if we our old parameter for
08:28 something like a chemical reaction
08:30 our order parameter our concentrations
08:33 are not conserved
08:34 now so we can change we can actually
08:36 increase the total concentration of
08:38 something
08:39 then we have equations of the first kind
08:42 called
08:42 model a this classification scheme
08:46 and if we are just moving around
08:48 concentrations
08:50 which i was actually not
08:53 without actually not taking any
08:55 particles for example out of the system
08:59 then we have these conservative
09:02 these conservatives so-called
09:04 conservative systems
09:06 which are described by equations of this
09:08 kind and they’re called
09:10 model e yeah and an example of this
09:14 functional f here is that we write this
09:17 as an
09:18 integral of some potential and
09:21 something that gives the diffusion term
09:24 once we take the derivative with respect
09:26 to the function of the derivative
09:28 with respect to phi now and this
09:31 potential here is also related then to a
09:35 local
09:35 force you know so that’s like a formal
09:38 analogy
09:39 and of course this the writing of such
09:42 kind of
09:42 systems comes actually from equilibrium
09:45 statistical physics
09:46 where this f is actually some free
09:48 energy
09:50 functions from generalized free energy
09:52 for example the fight to the floor
09:53 theory ginsburg land over a theory
09:57 and then you have an equilibrium system
09:59 and you want to know
10:00 how this equilibrium system evolves and
10:03 then you just take the functional
10:04 derivatives with respect to your fields
10:07 and then you know how your equilibrium
10:09 system your free energy function
10:12 your free energy that describes your
10:14 theory gives rise
10:15 to dynamics of the field now that comes
10:19 from equilibrium but it’s actually a
10:20 very inconvenient way for us to write
10:23 down
10:23 these kind of equations and uh
10:26 so i just wanted to tell you this
10:28 classification scheme
10:30 but in the following slides i write out
10:31 the equation directly
10:33 without going via dysfunctional
10:35 derivatives here
10:36 and as i said in this lecture we’ll
10:39 first
10:40 look at transitions between
10:43 non-equilibrium states uh
10:46 in systems that where this noise this
10:49 psi is not important yeah so that’s
10:52 noise for example is not important
10:54 if the temperature in the equilibrium
10:56 system the temperature
10:58 is zero or very small in a typical
11:01 biological system
11:03 noise typically is not important if you
11:06 have a very large
11:07 number of uh of uh
11:11 particles that contribute to a certain
11:13 process
11:14 yeah so we know first and the first step
11:17 we asked so
11:17 how can we go between different
11:20 non-equilibrium states
11:22 how can we switch to to say between
11:25 different kinds of order and
11:29 in the next step in the next lectures
11:30 we’ll ask okay what does this noise do
11:33 and of course i wouldn’t have a couple
11:36 of lectures
11:37 on this noise if it wouldn’t do very
11:40 interesting things
11:42 but for now we ignore the noise you know
11:44 and uh what we’ll do is also called
11:46 mean field theory just look at the
11:49 partial differential equations that are
11:52 drawn
11:53 that are driving these concentration
11:55 fields
11:56 phi of x t in the limit of a low
11:59 temperature or very high
12:01 particle numbers

slide 3

12:05 so
12:09 for the very first part of this lecture
12:12 i would like to even go to
12:14 even to an even simpler
12:17 framework and that is we don’t even
12:20 consider
12:21 space now we say the system is well
12:24 mixed
12:25 yeah and if the system is well mixed
12:28 then
12:29 we can uh neglect spatial derivatives
12:33 you know because the system is in
12:35 homogeneous state yeah and then
12:37 so this is considered like a chemical
12:39 and we’re always stirring the
12:41 these chemical reactions so that every
12:43 particle
12:44 every molecule very rapidly travels
12:48 through the entire system you know so
12:50 then we basically
12:52 have a homogeneous system where
12:54 concentrations do not depend on space
12:58 and if the concentrations do not depend
13:00 on space
13:01 then uh spatial derivatives become zero
13:05 yeah and these are the kind of systems
13:07 that we are looking at
13:09 formally again in this notation of
13:12 uh functional derivatives uh we can then
13:15 neglect
13:16 the spatial derivatives uh in this
13:18 functional here
13:20 and what we then get is a
13:23 differential equation of phi of t that
13:26 now does not depend on
13:28 space anymore and that is this time
13:30 evolution of this quantity
13:32 is just described by some function f
13:35 yeah and of course i could have just
13:37 started with this equation here without
13:38 the
13:39 function of derivatives and so on yeah
13:41 we just of course what we’ll
13:43 be looking at for for the first part of
13:45 this lecture
13:46 are non-linear differential equation or
13:48 nonlinear
13:50 dynamics and i’ll give you a brief
13:53 overview because it gives you
13:55 an insight not only on the
13:57 renormalization part
13:58 that we will do before christmas but
14:00 also to the second
14:02 part of this lecture we will look on how
14:04 spatial different spatial structures
14:06 can emerge so
14:10 we have these not with these non-linear
14:12 differential equations here so on the
14:14 left hand side we have the time
14:16 evolution of the scalar
14:18 yeah and on the right hand side we have
14:19 some function it’s a nonlinear function
14:21 that describes
14:22 this time evolution and uh
14:26 of course a typical example of a
14:28 non-linear system is always in biology
14:30 in biology everything is nonlinear and
14:36 very simple system you can look at is
14:39 for example
14:40 how the gene
14:44 uh interacts with itself a
14:48 self-activation
14:50 of a gene yeah and
14:54 if you have a gene i mentioned this
14:57 already now so you have a gene here
15:02 now that’s part of the dna and at the
15:05 beginning
15:06 of this gene also the dna is very long
15:08 we’re now looking at a very
15:10 short part of the total dna
15:13 that’s the gene here at the beginning
15:17 of this team there’s a promoter the
15:20 promoter
15:20 turns on or off this g and when this
15:24 promoter
15:25 turns on the gene then this gene
15:28 produces molecules
15:30 now actually via multiple steps
15:33 but in the end you have something that’s
15:36 called a protein
15:40 now of course i did this of course for
15:43 the ball it just of course it’s more
15:44 complicated than that yeah and this
15:47 protein what does this protein do
15:49 no it can degrade
15:52 but it can also do fancy things so for
15:55 this protein is produced and it swims
15:57 around in the cell
15:59 you know and we have these proteins now
16:01 here multiple copies of this because
16:02 this
16:03 gene keeps producing proteins and now we
16:06 can say that this protein
16:07 also decides whether this gene is on or
16:10 off
16:11 and what is a typical situation is that
16:13 this protein then binds
16:17 to this promoter to the start site of
16:20 this
16:21 and only if we have two of them together
16:25 we can start the gene
16:28 now we can start the gene and
16:31 so that means we need to find pairs
16:35 between these genes here but between
16:37 these proteins here
16:38 and if we have found a pair it can bind
16:41 and then
16:42 this starts producing proteins from the
16:44 gene again
16:45 which then again couple
16:49 back to itself so it’s a feedback and
16:53 uh typically the kind of equation that
16:56 you get from this
16:57 is from the
17:01 for the concentration of the numbers
17:04 of these proteins in the cell
17:08 is that you have one process that
17:10 describes
17:11 the activation of the gene itself
17:14 and this activation is non-linear so you
17:17 have many
17:18 different contributions y squared
17:22 divided by 1 plus y squared
17:25 this function here
17:28 is the activation part
17:33 it describes that you have to find pairs
17:36 of these genes now so that you look at
17:39 this here
17:40 the more pairs you have the more pairs
17:42 that’s the number of pairs that you can
17:44 you can build now the more pairs you can
17:47 build
17:49 the more likely you express this g here
17:52 you
17:52 turn it on but then we also have the
17:55 situation
17:56 that if we have too many of these
17:58 there’s a crowding effect on the dna
18:00 they can’t all bind
18:02 at the same time now so they have to
18:04 compete for binding
18:06 so they can’t if you have like a million
18:09 or like infinitely many of these copies
18:12 here
18:12 they all cannot bind simultaneously to
18:15 this region here because this is
18:17 only a finite amount of space yeah
18:20 and that’s why we have another part here
18:24 that saturates you know that means that
18:26 we
18:29 that we for very high values of this y
18:32 of this protein concentrations we cannot
18:34 get any better
18:36 and this is called the hill function and
18:38 the hill function typically
18:40 looks like something like this
18:49 now it’s clearly non-linear and then we
18:51 have the second term
18:53 now it describes that describes
18:56 degradation also and how how many
18:59 proteins or how many copies of these
19:01 proteins
19:02 we use at a given amount of time and
19:05 this is very simple because the more you
19:07 have the more you lose now so that’s
19:10 just
19:10 minus y and this is just as
19:14 we’ll look into detail into this
19:17 equation
19:17 later um and this is just an example of
19:21 how
19:21 in biological systems these non-linear
19:24 differential equations automatically
19:26 emerge
19:27 almost all the time already on the basic
19:31 building block of many biological
19:34 systems namely the expression
19:36 of a gene so
19:39 what can we not do with these nonlinear
19:42 equations
19:45 and i didn’t oh i had i had a transition
19:48 here i wouldn’t have okay so uh
19:52 so i had a fancy transition i wouldn’t
19:54 have needed to to write that

slide 4

19:56 uh okay so let’s move on what can we now
19:58 do
19:59 with such equations here so the
20:01 solutions
20:02 of these non-linear equations that live
20:05 in some
20:06 space in some configuration space
20:10 and in the space we move around you know
20:14 as the system evolves
20:16 now uh typically the the space of
20:20 all possible trajectories of all
20:22 possible solutions
20:23 such as the system is called face
20:25 portrait
20:26 and that we start
20:30 at some initial conditions and we
20:32 involve along
20:34 this trajectory here if we manage to
20:36 solve this equation
20:38 now there’s some points that are
20:40 particularly important
20:42 in this field namely these are fixed
20:44 points
20:46 and these fixed points are points where
20:49 the time derivative
20:50 is zero so once you are in these fixed
20:53 points
20:54 you cannot get out of that out again
20:56 because the time derivative
20:59 is zero now you stay there these are
21:02 called fixed points
21:03 and once we know the fixed points we
21:06 know already
21:08 a lot about the dynamics of a nonlinear
21:11 system
21:13 so now here i have the transition so on
21:15 the bottom
21:16 you can see how you can understand the
21:18 dynamics of nonlinear system
21:21 just by graphical analysis
21:24 so now on this diagram at the bottom
21:28 on the y on the x axis is just the
21:31 concentration
21:32 why not just any system we’re not just
21:35 looking at any general system i’ll use
21:37 the y’s for simplicity
21:39 for simplicity uh we have on the excess
21:41 if we have the concentration
21:42 y and on the
21:46 y axis we have what is ever is on the
21:49 right hand side
21:51 of our differential equation the times
21:53 the derivative
21:55 of y and now
21:58 we can of course plot this function we
22:00 can ask how does the right
22:02 hand side of our differential equation
22:04 depend
22:05 on the concentration y now and then we
22:08 get some function for example the ones
22:10 that i
22:10 plotted here and this function
22:15 will cross the zero line
22:18 now this function will be zero at
22:20 certain points
22:22 and wherever this function is zero this
22:25 is
22:25 where we have a fixed point now this is
22:28 where the
22:29 time derivative is 0
22:32 and then we can ask are these fixed
22:34 points stable
22:35 or are they unstable so once we’re in
22:37 there do we stay there forever
22:40 or is it enough if i give a little kick
22:42 to get out again
22:44 so how stable are these fixed points
22:46 once we’re in there
22:47 and that’s also something you can very
22:50 easily
22:52 see for example if you look at this next
22:53 point here
22:55 uh this fixed point here that y
22:58 dot or the time derivative is zero but
23:01 if you go a little bit to the right
23:03 then the time derivative becomes
23:05 negative
23:07 yeah so we go back into this point if we
23:10 go a little bit to the left
23:12 that the time derivative becomes
23:13 positive and we also get pushed back
23:17 into this point so
23:20 this point here is stable so that means
23:23 if we go to the right we get pushed back
23:25 in and if we go to the left we also get
23:28 pushed back
23:29 by the time derivative yeah and this is
23:33 just because
23:34 um the slope
23:38 here is negative so the
23:41 slope of this function of this time
23:44 derivative
23:45 now which is of course the same as
23:48 this here of this function the slope of
23:52 this function at the fixed point
23:54 tells us something about the stability
23:57 here is a so this is a stable fixed
23:59 point
23:59 now we always go back here this is an
24:02 unstable fixed point
24:03 so we go to the right and then the time
24:06 derivative
24:07 is positive so we get even further to
24:09 the right
24:10 we go to the left time derivative is
24:13 negative
24:14 and we get even further to the left
24:17 yeah so these fixed points are very
24:19 important and the stability of these
24:21 fixed points
24:22 tells us where our system will evolve
24:26 so just graphically you can see that if
24:28 i start here
24:30 with my system yeah you can you can just
24:33 graphically see
24:34 that my the dynamics will go into the
24:37 stable fixed point
24:38 and stay there now there are situations
24:42 so i said these fixed points are very
24:44 important that characterize
24:46 their stability the stability of these
24:49 fixed points characterizes
24:51 where our system will go carry towards
24:54 the dynamics of the system
24:56 and now what happens if we change
25:00 parameters now if these if we change
25:05 uh parameters then the number
25:08 or the kind of fixed point the stability
25:11 of this fixed point
25:12 can change and this is if this happens
25:16 uh that the number of this fixed point
25:18 or the stability
25:19 changes then are we talking about a
25:22 bifurcation
25:23 called a bifurcation and uh what is this
25:28 parameter
25:29 now so we call it r from now on so
25:31 there’s some parameter
25:33 that makes the number of fixed point
25:37 uh also this ability change that’s what
25:39 we call a control parameter and
25:42 typically
25:43 it’s related in many physical systems
25:45 related to
25:46 how far you’re actually out of thermal
25:48 equilibrium
25:50 so this r could be for example be
25:53 the val the difference in temperatures
25:56 between two boundaries
25:57 of the system
26:01 so these are bifurcations and
26:05 these bifurcations can be classified i
26:08 will now have a look at a few examples
26:10 of these
26:10 bifurcations so uh

slide 5

26:15 the simplest bifurcations or one of the
26:18 simplest bifurcations you can get
26:20 is if you consider nonlinear equations
26:23 of this kind here now so this is
26:26 for each bifurcation i show you the
26:29 simplest
26:31 differential equation that gives rise to
26:34 such a bifurcation
26:35 and the simplest equation is also often
26:38 called a normal form
26:41 so let’s have a look at this equation
26:45 now suppose that r is smaller than zero
26:48 so this equation here
26:49 so we plot the same thing as as on the
26:51 previous slide so on the
26:53 right hand on the y-axis we have the
26:56 time derivative
26:57 now which is just equal to whatever is
26:59 on the right-hand side
27:01 and on the x-axis we have our
27:03 concentrations
27:05 now if this r is negative now then we
27:08 just have a simple
27:09 parabola that is shifted that is shifted
27:12 down
27:13 now and if you have that we can do the
27:15 same argument as
27:16 previously so we are at this fixed point
27:19 we go to the right
27:20 and then the time derivative gets
27:22 negative so you would really push back
27:25 now into this fixed point so this fixed
27:27 point is stable
27:29 and then on the right hand side we have
27:30 another fixed point which is
27:32 unstable in the middle
27:36 if r is exactly equal to zero
27:39 then we have a parabola well it’s not
27:42 hard to see on this parabola we have a
27:45 weird fixed point here at the bottom
27:47 now we’re not really sure whether it’s
27:49 unstable or not
27:50 it’s at the just at the boundary between
27:53 stable and unstable
27:55 so we go here it’s stable from the right
27:57 hand side
27:58 and unstable from the left hand side the
28:01 typical notation is
28:02 that filled circles of this
28:06 are denotes stable fixed points and
28:08 these open
28:10 or white circles denote
28:13 unstable fixed points and then here the
28:15 idea is that this
28:16 fixed point is stable from the left and
28:18 unstable
28:19 to the right now and then we set r to
28:22 positive values
28:23 then we don’t have any fixed point at
28:25 all and our system
28:27 will just go to infinity now the time
28:30 derivative
28:31 is always positive it will just go to
28:33 infinity and there are no fixed points
28:36 what we can now do is we can plot
28:40 the location of these fixed points that
28:44 i’ve given you now for three specific
28:46 values
28:47 of this r this parameter r
28:50 we can plot the location of these fixed
28:52 points
28:53 continuously as a function of r and
28:56 that’s
28:57 depicted in the so-called bifurcation
28:59 diagram
29:00 now and i’m showing you here the
29:02 bifurcation diagram
29:04 of non-linear differential equation that
29:06 you see on top here
29:08 and in this case you can see
29:12 that the fixed points location of fixed
29:14 points were here
29:15 we have a stable fixed point negative
29:18 values
29:19 and then an unstable fixed point at
29:21 positive values
29:23 and then as we go increase our values of
29:26 r these two fixed points merge
29:29 and we end up on the right hand side
29:31 with a state
29:32 where we don’t have any fixed point at
29:34 all and we just go
29:36 to a very high values of the
29:39 concentration
29:40 of y and so this is how to read these
29:43 bifurcation diagrams

slide 6

29:45 and just to give you an example just to
29:49 come back
29:50 to the self-activation of this gene now
29:53 this example
29:54 here we have the nonlinear differential
29:57 equation again
29:58 protein concentrations
30:01 activation term this non-linear term
30:05 that’s actually something you can
30:06 calculate in a longer calculation
30:08 but it’s already clear from the from
30:11 these pictures that there’s something
30:12 non-linear
30:13 coming up and the degradation term
30:16 you know and then we plot both terms
30:19 separately
30:20 and we get something like this here this
30:22 is the activation term
30:24 this is the degradation term and uh
30:27 if we sum them up we get something
30:30 that looks like what we have here in the
30:32 middle
30:33 you know where we then is basically the
30:36 same plot i showed you in the last slide
30:38 we plot the right-hand side as a
30:40 function of y
30:41 and then we see that for small values
30:45 of r here we have just one fixed point
30:50 and as we increase r this
30:53 non-linear term will become more and
30:55 more important
30:57 now and we start
31:00 having intersections with the x-axis so
31:04 with zero
31:06 and if we have a very large and strong
31:08 activation
31:09 very strong feedback on itself we have
31:12 multiple fixed points here
31:15 the right-hand side is the bifurcation
31:17 diagram
31:18 this bifurcation diagram so here’s the
31:22 fixed point and uh this bifurcation
31:25 diagram looks
31:26 as as follows yeah so as you go have
31:29 very
31:29 for very low values now it’s just a
31:33 translation of these red
31:34 points here uh from from the previous
31:38 picture
31:38 for very low values you have a stable
31:40 fixed point here
31:42 at zero and then suddenly this
31:45 non-linearity or this
31:47 sigmoidal curve kicks in becomes
31:50 important
31:51 and you start intersecting with zero
31:54 yeah and that happens here at this
31:57 what’s called a bifurcation point
31:59 and then this bifurcate and if you go
32:01 past this bifurcation point
32:03 you go to a state you have a stable
32:05 state here
32:07 and an unstable state here and a stable
32:10 state at the bottom
32:12 just remains there all the time at zero
32:17 so this is an example of a saddle node
32:19 bifurcation so it has this typical
32:21 signature here of the saddle node
32:23 bifurcation a little bit more
32:24 complicated
32:25 yeah but what you see here is what
32:28 happens
32:29 if you turn on the self activation if
32:32 you turn on the non-linear term
32:35 you go to a regime you go through this
32:37 bifurcation
32:38 here where your system has two stable
32:42 fixed points you know two stable fixed
32:45 points
32:46 are here and here and if you have two
32:50 stable fixed points that are separated
32:52 by an eight unstable one then you have a
32:54 switch
32:55 so this is just an example of how
32:59 biological systems in this type of gene
33:02 can make use of this nor these
33:04 non-linear effects
33:06 that they get for example here by having
33:09 clusters or little pairs you’re
33:11 requiring to have pairs
33:12 of proteins to bind to the starting
33:14 region of the gene
33:15 how they can make use of this nonlinear
33:17 in fact
33:19 to in this case build a switch
33:22 and with this switch if you have a
33:24 switch you can it’s like a bit
33:26 you can actually store memory in a
33:29 stable way
33:30 and uh this is one of the simplest ways
33:33 that
33:34 uh biological systems or that cell can
33:36 store
33:37 information okay so let’s go on i’ll

slide 7

33:40 just show you some other
33:42 bifurcations now so we have here
33:45 certainly different differential
33:46 equation now with
33:48 r times y plus y to the
33:52 power of 3 and then if you just look at
33:55 the right hand side and you just do this
33:57 graphical analysis
33:58 you will see that for small values of r
34:00 you have a single fixed point
34:02 single intersection and then if you look
34:05 at the slope
34:07 now you can see that this is actually
34:09 stable as you go to the right
34:11 and the derivative becomes negative so
34:13 you get
34:14 pushed back you go to the left and the
34:16 derivative becomes positive it always
34:18 pushes you in the opposite direction
34:20 so one stable fixed point so
34:23 if r is exactly equal to zero you’re
34:26 still stable but you’re in this real
34:28 state
34:29 where you have um a flat
34:33 as your function goes to tangential
34:37 to the zero axis to the y-axis
34:40 and um that oh sorry
34:45 i activate it and that reminds us a
34:48 little bit
34:49 to second order phase transitions you
34:52 know so that you get
34:53 tangentials because you’re in a state
34:54 where you whether you don’t really know
34:56 where they should go left or right
34:59 now so that this function is flat you
35:01 can go
35:02 you can go left and right but uh it’s
35:05 not really punished
35:06 so you can you can you can go left here
35:09 but because this function is rather flat
35:11 now you can
35:12 stay there for a long time you go right
35:14 and the function is very flat
35:16 now it’s tangential you can also stay
35:18 there
35:19 and that’s probably what you know from
35:21 the potential
35:22 in the isaac mode for example second
35:25 order phase transitions
35:26 where also at the critical point the
35:29 potential becomes flat
35:30 and then fluctuations to the left and to
35:33 the right and spins
35:35 are not punished anymore and you get
35:36 these long range
35:38 correlations in in the fluctuations and
35:41 all these weird effects of criticality
35:44 now if you increase r further
35:48 then you get something like this here
35:50 you’ve got two fixed points
35:55 you get two fixed points two stable ones
35:57 and an unstable one
35:58 in the middle and the bifurcation
36:00 diagram looks like this here
36:02 uh so for low values of r you have
36:05 one stable fixed point at zero
36:09 and then as r increases beyond
36:12 uh critical value beyond r equals zero
36:16 you get this branching into two stable
36:19 states
36:20 that are separated by an unstable state
36:23 and if you now compare again to the
36:25 ising model
36:27 uh this is exactly how the magnetization
36:29 looks like
36:30 as a function of the temperature of the
36:33 inverse temperature
36:34 this looks like an icing mold where you
36:36 lower the temperature
36:42 so this was a so-called super critical
36:44 pitch for
36:45 bifurcation and uh there’s a super
36:48 critical pitch book bifurcation that’s
36:50 also a subcritical
36:52 pitchfork bifurcation now that looks um

slide 8

36:55 like this here and there is an error in
36:58 this
36:59 formula let me just check
37:06 um
37:08 here sorry
37:11 there’s a minus sign
37:15 it should be minus here
37:19 now for this to make sense for these uh
37:21 cross made sense needs to be a minus
37:23 and now we have the same thing but with
37:25 a plus
37:27 yeah and if we have this plus then we
37:29 just turn around
37:30 the diagrams that we have the previous
37:33 slide
37:34 so for low values for negative values of
37:38 control parameter r uh we get three
37:40 fixed points
37:42 one stable fixed point in the middle and
37:44 two unstable fixed point
37:46 points at boundaries as we increase
37:49 r we have one unstable fixed point
37:53 at uh y equals zero and
37:56 uh this fixed point stays unstable
38:00 as we increase the value of r
38:04 so here’s the bifurcation diagram
38:07 so we start with low values of r where
38:09 you have a stable
38:11 fixed point at the concentration zero
38:15 and two unstable fixed points
38:19 around that so because they’re unstable
38:22 you have to go this way
38:24 as well and then as you increase
38:28 this value of r you go to a state
38:31 where your fixed points the stable fixed
38:34 point you’ve been in
38:35 suddenly becomes unstable yeah and
38:40 what you have here is now that if you go
38:43 here so here you stay at zero
38:45 you stay at zero all the time and then
38:48 you go to this state and then you don’t
38:50 go to something small
38:51 but you immediately to go go to
38:53 something very large to infinity if
38:56 there’s no other thing that
38:57 stops you from doing that and that’s a
39:00 sub critical bifurcation
39:01 where it’s because you have this uh this
39:04 uh
39:06 this discontinuity in the state of your
39:08 system so here it was zero
39:11 and suddenly it becomes very something
39:12 very large
39:14 if you compare that to a supercritical
39:16 bifurcation
39:18 our state was zero for small values of r
39:21 and then continuously increased so if
39:24 this was a
39:25 was resembling a second order phase
39:27 position
39:28 now this is resembling a first order
39:30 phase transition
39:31 right in the isaac model for example if
39:34 you change the
39:35 magnetic fields at low temperatures
39:39 so so what’s so i’m making this
39:42 correspondence to
39:44 ising models and uh equilibrium systems
39:47 and phase transitions here
39:49 so what’s the difference between a
39:50 bifurcation and a face
39:52 uh transition so biovocation
39:56 bifurcations actually resemble phase
39:58 transitions
39:59 in specific cases namely when
40:05 our when this here is actually a free
40:09 energy yeah and that’s
40:10 that’s that’s the beautiful thing about
40:12 this writing uh these nonlinear
40:14 equations in this
40:15 specific form yeah so if this here is
40:18 actually a free energy
40:20 like the vinsmoke lambda or free energy
40:22 function for example that describes
40:23 things like the ising model
40:25 then the bifurcations correspond our
40:29 generalization of phase transitions
40:33 now you have many bifurcations
40:36 mainly possible bifurcations including
40:40 bifurcations that have
40:42 imaginary components so that give rise
40:45 to
40:45 imaginary components then that means
40:48 that you have
40:49 oscillations in time so that’s that’s
40:52 also something that
40:53 that you can have in these bifurcations
40:55 and but we’ll not be dealing with that
40:57 i’ll show you one more kind of
40:59 bifurcation and that’s
41:01 a trans critical modification i’m
41:03 showing you that because it’s relevant
41:04 for epidemics
41:06 and for the things that we’ll be doing
41:07 before christmas
41:09 yeah and because we’re now probably
41:11 going into lockdown
41:13 sooner sooner than later in jason and
41:15 also have
41:16 spent the last rest of the time for
41:18 christmas working on
41:20 epidemic models i’ll explain to you
41:22 renormalization on
41:23 epidemic models yeah and this is an
41:26 example of an epidemic model i’ll show
41:28 you in the next slide
41:29 why this is the case this is just again
41:31 now the simplest equation that gives you
41:34 this kind of behavior so r times y minus
41:37 y squared but now you do the usual
41:40 graphical analysis
41:42 and what you see is that you have this
41:44 inverted
41:45 parabola and if r is smaller than zero
41:49 you’re gonna have something like this
41:50 here
41:51 and if you increase r yeah
41:54 then you move uh to a single fixed point
41:58 and then you go to a stable fixed point
42:01 at positive values of r
42:04 so and this is the bifurcation diagram
42:06 here at the bottom
42:08 i’ll show you how to in the next slide
42:10 i’ll give you an example
42:11 we have a stable branch for low values
42:14 of r
42:15 and then you go and an unstable branch
42:18 here for negative values of
42:19 y because stable at zero at negative
42:22 and unstable negative values and then
42:25 you flip
42:25 things around and the unstable branch
42:29 the zero point becomes unstable and
42:33 this diagonal line here this linear
42:36 state
42:36 becomes stable that’s called a
42:38 trans-critical bifurcation and you just
42:41 flip things around basically
42:44 now let’s have a little look at such an
42:47 example of a trans-critical
42:49 bifurcation now so suppose

slide 10

42:53 you have a disease now let’s not give it
42:56 a name
42:57 so last year last year i gave a lecture
43:00 and i introduced disease models a few
43:03 series of disease models
43:05 and it was february last year and
43:08 these disease models at this point i
43:11 called it the rouhan
43:13 virus because at this point of the wuhan
43:16 model because at this point
43:17 the pandemic was restricted to this one
43:19 city in china
43:21 but now it’s a little bit more general
43:24 and that’s now we call it i don’t know
43:26 the world iris or whatever now so this
43:30 model looks very similar simple
43:32 now so you have two kinds of people and
43:34 also the the infected ones
43:36 and the susceptible ones not infected
43:39 ones
43:40 they carry the disease they carry the
43:41 virus and the susceptible ones
43:44 they are healthy but they can catch the
43:48 virus
43:50 it’s the simplest disease model you can
43:52 think about it’s called also called the
43:53 because you have these two uh two
43:56 letters called the s
43:57 i model or contact process now
44:01 we can write down some simple chemical
44:04 reactions
44:05 some pseudo chemical so if an
44:08 infected person meets a susceptible
44:10 person or a healthy person
44:12 then with a rate lambda the the
44:15 susceptible person
44:16 turns into another infected person
44:20 and we have two infected persons at the
44:23 end of this
44:24 reaction yeah and then
44:28 the second thing that can happen is that
44:29 an affected person at some point
44:31 recovers
44:32 you know and if you recover you turn an
44:35 infected person
44:37 back to a susceptible one now that’s the
44:40 simplest thing you can imagine in terms
44:42 of disease spreading
44:43 and now we can have a simple look at uh
44:46 how we
44:47 understand the non-linear dynamics of
44:50 the system
44:54 so first we just write down differential
44:57 equations
44:58 what is the time derivative
45:02 of the concentration of these i people
45:06 now we can write down this time
45:07 derivative so this the number of
45:09 infected people
45:11 increases with the rate lambda
45:14 and this rate of increase is a
45:17 proportional to the probability that is
45:19 a susceptible person meets
45:22 gets in touch with an infected person
45:25 and the more affected and the more
45:27 susceptible people we have
45:29 the higher is the spreading rate so this
45:32 is
45:32 proportional to s times i
45:36 and then an infected person can
45:39 turn back into a susceptible one
45:43 so that means we have minus s i
45:47 minus mu i
45:50 we can write down a similar equation for
45:52 the susceptible people
45:54 d over dts and that’s just the reverse
45:58 now so the the negative of this so
46:01 we lose so it’s a infected people by
46:05 infections number times as i times
46:08 s times i and plus
46:12 whenever an infected people uh
46:15 recovers
46:19 we get another susceptible one
46:22 and uh what we also say is that’s uh
46:25 such a simplification this is an
46:28 important simplification
46:29 is that the total number of people
46:33 stays constant like we say this is a
46:37 total concentration of both gas content
46:39 so this hectic turns into susceptible a
46:42 susceptible tends to affect it
46:43 but actually people don’t die from the
46:46 disease
46:47 so the number of people that we have
46:50 remains constant
46:52 now we can plug this condition in
46:56 then we get d over dti
46:59 is equal to lambda i i minus 1
47:03 now we just plug this in minus
47:08 ui and then we can get the fixed points
47:12 by just setting this to zero
47:15 the fixed points are given by i
47:18 times lambda 1 minus i
47:23 minus mu is equal
47:27 to zero so this is not the imaginary i
47:29 of course that is just the infected
47:32 and uh so this is the condition if we
47:34 set the left-hand side of these
47:35 equations to zero
47:37 and then we get a condition for the
47:38 fixed point and then we can solve this
47:41 and say okay i one is zero
47:45 the first pixel fixed point is at zero
47:47 so we can solve this equation by setting
47:49 i to zero
47:50 we can solve this equation also by
47:53 setting
47:54 i to lambda minus
47:57 u over lambda now that’s another
48:00 solution
48:02 which is equal to one minus mu over
48:06 lambda this tells us already that this
48:09 mu over lambda is something important
48:11 the ratio between the time scales the
48:13 rates of these processes
48:15 is something important because it pops
48:17 up here
48:18 in the fixed points as a ratio
48:22 and now now we say if you evaluate now
48:25 the right hand side
48:26 of this equation at the fixed point to
48:28 get the stability
48:30 yeah so the time derivative with of
48:33 this right-hand side that’s called f
48:36 and that is just given by lambda
48:40 1 minus 2i minus
48:43 mu and
48:46 now we evaluate this time derivative
48:49 this derivative
48:50 at the fixed point so the first fixed
48:53 point
48:54 is zero
48:57 and at this fixed point we have lambda
49:01 minus mu where we plug that in and the
49:03 second fixed point is
49:05 f prime of 1 minus nu over lambda
49:09 and then we have that this is
49:13 u minus number
49:16 so this looks a little bit symmetrical
49:18 right and this reminds of
49:20 us of the this transcritical bifurcation
49:23 that we had
49:24 and uh if we plot things then we see
49:28 that that’s actually what’s happening
49:30 now we plot the fixed points
49:31 now this is ice sorry
49:35 i star as a function
49:39 of mu over lambda
49:42 all right so then we have the staple
49:44 fixed point
49:46 so we see here that there’s that the
49:48 signs of this fixed point
49:51 whether they’re stable or not that
49:53 depends
49:54 on whether this what is the whether mu
49:58 is larger than lambda or not
50:01 so something is happening here at one
50:04 and now we plot this fixed point so one
50:06 is at zero
50:07 and for low values if uh
50:11 if lambda is larger than mu yeah
50:14 then this fixed point here is unstable
50:19 that’s the dashed line and the other
50:22 fixed point
50:23 just has the opposite stability it’s
50:26 stable
50:27 goes like this and then
50:31 if at that lambda here at mu over lambda
50:35 equals to one we have this change where
50:38 now
50:40 this zero fixed point this one here
50:44 becomes stable and the other fixed point
50:49 becomes unstable so this simple disease
50:52 model shows a transcritical
50:54 bifurcation and if we now take into
50:58 account fluctuations that we will
51:00 do that before christmas we’ll see that
51:02 this is actually
51:05 that this model is actually one of the
51:08 fundamental model
51:09 to understand criticality and
51:12 non-equilibrium systems so this
51:14 simple model describes the large class
51:16 of
51:18 critical behaviors in non-equilibrium
51:20 system and we’ll see that in the
51:22 following lectures
51:24 so this was just uh briefly a discussion
51:27 of what can happen
51:29 if homogeneous states change if you
51:32 don’t have states
51:34 and uh so that was something that they
51:37 basically the foundations of nonlinear
51:39 dynamics

slide 11

51:40 many of you will already have heard of
51:41 that and
51:43 of course non-equilibrium systems
51:46 have this capacity that they’re able to
51:49 produce
51:50 very complex uh structures so if you
51:52 think you’re just in space i first think
51:54 for example
51:54 about biological system think about a
51:57 cell and all of this stuff that is
51:58 highly organized in the cell
52:01 yeah so in this second part of this
52:04 lecture we will now want to understand
52:06 if we not only have transitions between
52:08 homogeneous states
52:09 but can we also have transitions between
52:11 homogeneous states
52:13 yeah so where that have no spatial
52:15 structure where all for example
52:17 all arrows or all spins point in the
52:20 same direction
52:21 and states where we actually have um
52:24 a spatial pattern or a spatial structure
52:28 and a nice example so one of my favorite
52:31 examples
52:32 is actually you can see here on the
52:33 surface of jupiter
52:35 and you can see
52:38 now a satellite image of jupiter here
52:41 and what you see is that you have here
52:45 these stripes
52:47 on the surface of jupiter now you have
52:49 stripes
52:50 of different color of different kinds
52:53 and what’s actually happening here is
52:55 that you have
52:56 a balance between uh convective
53:00 processes so
53:02 so gas that is that comes from a jupiter
53:05 from the
53:06 core of jupiter and that rises to the
53:07 surface and then goes back
53:10 and you have shear flow also where these
53:13 actually if you look at jupiter as a
53:14 movie
53:15 after that you will see that some of
53:18 these drives travel in the left
53:19 direction
53:20 and others travel in the right direction
53:22 it’s very it’s very very cool actually
53:24 and the reason for this is that it is a
53:27 non-equilibrium system
53:29 and once that one that fits very well
53:32 into our definition that we had in the
53:34 first lecture
53:35 namely the system is coupled to
53:37 different bars
53:38 and so this jupiter is hot inside
53:43 and cold outside so on the outside and
53:46 we have space
53:47 space and that’s very cold and inside
53:49 jupiter is very hot
53:50 yeah and if you do that you have
53:52 something hot and something cold
53:55 now you know that frog maybe from your
53:56 room then you get conductive flow so the
53:59 air goes up
54:00 cools down goes up cools down
54:04 gets heated up cools down and so on you
54:07 get these convective flows
54:08 and that generates these patterns on
54:12 jupiter
54:13 and the origin of these patterns of this
54:15 conductive flow
54:16 is that you have this incompatible bath
54:19 the cold bath
54:20 or the cold boundary or the hot boundary
54:23 at the bottom
54:24 and the code boundary at the top and
54:26 that gives rise to
54:28 spatial and dynamical structures that
54:30 look very interesting

slide 12

54:35 so now we go back to our little
54:38 uh a little general functional
54:41 definition of spatial
54:43 uh launch voice system again we look uh
54:46 we ignore the noise again
54:48 and again also if you if you’re not
54:50 familiar if you’re not very happy with
54:52 these functional
54:54 derivatives uh i always write down the
54:56 specific
54:57 equations that we’re actually studying
54:59 at the following but this was the
55:00 general framework that we studied and
55:03 that we introduced
55:04 that incorporates both the conservative
55:06 and the non-conservative models the
55:08 model a
55:08 and b and we suppose that there’s some
55:11 parameter
55:13 r here
55:16 that describes how our system goes out
55:20 of equilibrium
55:21 also that typically describes
55:24 a transition a control parameter that
55:27 was previously bifurcation
55:29 but that now describes a state where we
55:31 go from
55:32 a spatial homogeneous spatially
55:34 homogeneous
55:35 solution spatially homogeneous system to
55:38 a system
55:39 that is spatially structured now and
55:42 that’s also what’s here on the right
55:43 hand side
55:44 yeah and you have this parameter r and
55:47 if you increase
55:49 this parameter r and you ask
55:52 whether or not you have a spatial
55:55 pattern
55:56 then you want to understand this
55:58 transition between
56:00 the stage where you don’t have any
56:02 pattern no the homogeneous day the
56:04 boring state
56:05 and the stage where your system is
56:07 structured and it has a characteristic
56:10 wavelength
56:11 and so on and this parameter we call
56:14 r again and
56:18 an example of such a system now as you
56:21 can see here so if we
56:22 plug in some values for this function
56:25 here
56:25 we get and partial differential
56:27 equations where we have a time
56:29 derivative
56:30 here again on the on the left hand side
56:33 and we have some non-linear terms
56:35 here on the left hand side but we also
56:39 have and this
56:40 actually looks like something that we’ve
56:41 seen i probably was the supercritical
56:45 bifurcation but we also have
56:48 spatial derivatives of any order so here
56:51 we have
56:52 the second spatial derivative like
56:54 diffusion
56:55 term and we have a fourth order
56:58 spatial derivatives of the fourth
57:00 spatial derivative
57:02 with respect to space now so this is an
57:05 example of the kind of systems
57:08 that describe spatially extended
57:11 systems if we neglect noise
57:18 so how do we now study this kind of
57:20 systems
57:21 yeah so how do we study that the idea
57:24 is that like in many
57:28 cases in physics that we look very
57:31 closely
57:32 at these points here we look very
57:35 closely at the point
57:37 when we see a pattern emerge for the
57:40 very first time
57:42 now we go to the threshold value to this
57:44 bifurcation point
57:46 and the idea is that we
57:49 linearize around that suppose now you
57:52 have a system yeah so
57:54 think back about uh our original our
57:57 lecture from last time there we had
57:59 rotational
58:00 invariance now so we have rotation and
58:03 variance so we’re pointing in different
58:05 directions
58:06 and then we ask how can we break
58:08 rotational
58:09 variance how can we make the system
58:13 globally point into one direction
58:16 and now we ask a similar question so we
58:18 start with a system that is
58:21 translationally invariant so that it’s
58:24 homogeneous in space so we move it
58:26 around

slide 13

58:27 now from here to there and it doesn’t
58:29 change and that means it’s a homogeneous
58:32 in space now there’s no structure in it
58:36 now how can we now break translational
58:39 invariance that’s a similar question to
58:42 what we had about the
58:44 rotational invariance so how
58:47 and under which condition is a
58:49 translational invariant is broken
58:50 and the idea is that we start
58:54 with a homogeneous solution yeah
58:58 let’s go back here we start with a
59:00 solution
59:01 where we have no pattern i like this
59:04 branch here
59:05 and the y-axis is something like that
59:07 quantifies a pattern
59:09 now so we have here we have this
59:10 homogeneous state
59:12 and then we look at small perturbations
59:14 around that and if we say
59:16 we hope that if we understand small
59:19 perturbations around this homogeneous
59:21 state
59:22 then we can actually learn something
59:24 about the real
59:25 macroscopic states that evolve
59:29 and that works
59:33 this idea works if we have something
59:36 like here
59:37 you know if we have something like here
59:39 this picture
59:40 where a pattern continuously emerge
59:43 emerges so we exchange some control
59:46 parameter
59:48 and then if we change this for control
59:50 parameter we first get a very weak
59:52 pattern
59:53 we get a stronger pattern and even
59:55 stronger pattern and so on
59:57 so this this bifurcation of how we get a
60:00 pattern is continuous
60:01 and one example here is the
60:03 supercritical
60:05 pitchfork bifurcation that is depicted
60:07 here
60:08 or that is resembled in a homogeneous
60:11 system by this kind of bifurcation
60:13 so how does this work also what we do is
60:16 we say
60:17 that our state that has a spatial
60:20 dependence and a time dependence
60:23 if gear is given by some homogeneous
60:25 state now we say the system is stable
60:28 in some boring homogeneous state
60:32 and then we have a little perturbation
60:34 around it
60:36 and now we ask whether this perturbation
60:38 will grow
60:39 or not and we’re not asking just about
60:43 any percolation we make a specific
60:47 answer for these perturbations you know
60:51 to make it answers
60:55 oh sorry wrong color
61:00 bring ons us
61:03 for the growth
61:08 of periodic perturbations
61:15 let me see if i have that oh i don’t
61:18 have two answers already here
61:19 okay so okay great so here we see
61:22 i don’t have to write that down so we
61:25 make it answers for
61:26 periodic perturbations yeah and this
61:29 answers
61:31 looks as following that we say that our
61:33 little perturbation here
61:35 that we with a linear order because our
61:38 little perturbation has two components
61:41 one component describes
61:45 the time evolution of our perturbation
61:49 now and depending that has some rate
61:51 here some pre-factor sigma q
61:53 and whether the sigma q is positive or
61:56 negative
61:57 tells us whether this perturbation will
61:59 grow or shrink
62:02 and then we ask here then we have here
62:04 this
62:05 imaginary part now this can also be an
62:07 imaginary
62:08 this is this this complex part
62:11 where we have essentially a periodic
62:14 pattern
62:15 now that’s the complex representation of
62:17 a periodic
62:18 pattern and here we have a pattern that
62:21 has
62:22 a wave vector q
62:26 and now we ask if we make this answer
62:28 for some
62:29 values of q
62:32 now for some values of q we have this
62:34 periodic perturbation
62:36 around this homogeneous state does it
62:38 grow
62:39 or does it not grow and we ask this
62:41 question
62:42 for every value of this wavelength
62:46 with which we perturb the homogeneous
62:48 states

slide 14

62:51 yeah and then several things can happen
62:54 that’s another
62:55 transition uh now several uh things can
62:58 happen
62:58 so if the real part of the sigma that is
63:02 a function of q
63:03 in the end is negative
63:06 yeah then this homogeneous state is
63:10 stable
63:10 we say it’s linearly stable and
63:13 and because this homogeneous state is
63:15 stable we don’t expect to see any
63:17 spatial structure
63:19 to emerge now is this phi
63:24 if this real part of
63:27 sigma q is positive yeah
63:31 then this term here grows and grows and
63:33 grows
63:35 now then if for some value of q this is
63:38 positive
63:39 then we get a pattern because our
63:42 periodic activation
63:43 grows and constantly becomes bigger and
63:45 bigger
63:47 now if you look at this here we can have
63:49 this suppose we get this
63:51 sigma of q we get the rate of growth for
63:54 each vector for each wave vector q
63:56 here then we can plot this as a function
64:00 of our control parameter
64:03 and what you sometimes see is that this
64:06 function
64:07 has some function and it’s always
64:09 negative
64:10 here and then at some value of r
64:14 of this control parameter we start
64:17 intersecting
64:18 with this zero point
64:22 and one wave vector
64:25 begins growing while the others are
64:27 still suppressed
64:28 and then if you increase r further
64:32 then uh you have a broader
64:35 number of a broader range of wave
64:38 vectors
64:39 that that start growing
64:42 and this wave vector qc of this
64:45 wavelength the corresponding wavelength
64:49 of our perturbation that for the first
64:51 time
64:53 becomes positive yeah in this
64:55 bifurcation
64:56 when we start seeing a pattern this we
64:59 say
65:00 gives us the wavelength of the final
65:02 pattern that gives us the length
65:04 of the final pattern and of course there
65:08 for this to work so we need to be very
65:10 optimistic are we
65:12 linear a lot we linearize you know we
65:15 say okay so this is something like this
65:17 and then we make this answer
65:22 we make an answers that uh and then
65:25 we say okay so this unlocks although
65:27 it’s very small it describes whatever is
65:29 happening
65:30 on very large scales on on
65:33 even if we waited for a very long time
65:36 no
65:36 and this works very often but a
65:38 situation where it doesn’t work
65:40 as you can see from here is where this
65:43 bifurcation is actually not continuous
65:45 but discrete
65:46 now for example like in super critical
65:49 in subcritical
65:50 bifurcations where you suddenly jump to
65:52 a pattern forming state
65:53 then this linear stability or
65:55 instability analysis
65:56 does not work
66:00 so yeah yes
66:03 is it in the chat at all okay
66:09 let’s see how we can see the chat here
66:18 ah no you know for some reason
66:22 for some reason i can’t see the chat can
66:25 you tell me the questions
66:34 awesome so what kind of perturbation you
66:36 also said you hope
66:37 i i saw i hope so i have most cancelling
66:40 headphones so i hope i have uh
66:44 the question correctly so there’s a
66:46 question of what kind of perturbation do
66:47 you put into this
66:48 state so that so you hope that it
66:52 doesn’t matter
66:53 now but the simplest answers you can
66:55 make
66:57 is just what i’ve shown here yeah could
67:00 have shown you
67:01 of course you can make different
67:02 perturbations now that are more
67:04 complicated but then the mathematics
67:06 gets too complicated and of course
67:08 what’s happened here is i’ll show you
67:10 now a
67:10 full calculation of this i’ll show you
67:12 an example of course what happens here
67:14 is
67:15 what you could do is you just go to what
67:18 you’re doing here is to go to various
67:19 space
67:20 yeah this perturbation that wrote down
67:22 and down here is something like the
67:23 fourier transform
67:25 of your perturbation yeah and then you
67:28 say that
67:28 one wave vector is the important one so
67:31 that these wave vectors don’t really mix
67:34 so that’s that’s the idea behind that
67:36 now but the
67:37 idea is in linear stability and
67:38 stability analysis and that’s why
67:40 they’re
67:41 you always have to check it with other
67:43 methods uh
67:44 is that uh of course the kind of
67:48 perturbation if the kind of
67:49 perturbations that you make
67:51 here would be important for the end
67:53 result then this whole thing wouldn’t
67:55 work
67:56 and it only works of course because you
67:58 are allowed to linearize
68:00 and uh because you assume that these
68:02 different
68:03 q values don’t interact with each other
68:07 in some some some complex way
68:10 i don’t know if this was the question is
68:12 basically you put in some
68:13 some very weak uh periodic perturbation
68:18 you know so you can have this unzots
68:21 which is essentially like a sine or
68:22 cosine
68:24 and see if this answer grows
68:27 or shrinks and that then tells you
68:31 how your what in the linear regime the
68:34 linear approximation
68:36 uh how your system reacts to
68:38 perturbations
68:40 and then you assume that if you wait
68:43 long enough you look macroscopically at
68:45 your pattern
68:46 like a jupiter that the wavelength that
68:49 grows strongest
68:52 once you go through this bifurcation
68:53 here this wavelength that grows
68:55 strongest is the one that will actually
68:58 then dominate
68:59 also in the long term
69:02 yeah so that’s what you think it works
69:04 it works very well yeah so
69:06 but only under constraints under certain
69:08 conditions
69:11 um before i show you a specific example
69:14 there of course is there’s a whole
69:16 classification
69:17 of these instabilities of how you can
69:20 generate a pattern
69:21 and that also depends it all depends on
69:25 how our sigma of q this function sigma
69:28 of q
69:31 looks like as you increase this r
69:34 parameter
69:35 that drives us from the homogeneous
69:37 state to a pattern state
69:39 now for example if you have there’s a
69:41 type one instability that i just showed
69:43 you
69:44 and uh this is so in this type 1
69:47 instability
69:48 you have this parabola like shaped where
69:51 you have a maximum
69:53 at a finite wavelength or wave factor to
69:56 finite wave vector
69:57 and there’s one specific wave vector
70:01 that will start growing uh
70:06 in a very well defined way now so so
70:08 here you have one specific
70:10 finite wave vector that will
70:14 dominate this process and that’s called
70:16 type 1 and stability
70:18 there’s a type 2 instability as well and
70:20 that’s a little bit complicated so let’s
70:22 let’s
70:23 let’s maybe first start with the type
70:24 well this is a type 3 instability
70:27 and there also you have a wave vector
70:30 that has the dominant
70:32 growth well that has the
70:35 the maximum of this function sigma of q
70:39 uh in this case is at q equals zero
70:44 also wave vector zero and wave x zero
70:48 means that you have a very long
70:49 wavelength
70:50 and that means your whole system is
70:52 essentially homogeneous
70:53 so instabilities of types type three
70:59 gives you situations where actually we
71:01 go from homogeneous state
71:03 to another homogeneous state the reason
71:06 why these
71:06 instabilities are important is that you
71:09 can also have situations
71:11 where the sigma of q has uh
71:14 an imaginary part and if the sigma of q
71:18 has an
71:18 imaginary part then this first part here
71:22 also
71:22 describes an oscillation yeah then you
71:25 have an oscillation not only in space
71:27 but also in time so you can have also
71:31 instabilities where you actually go from
71:33 a homogeneous state
71:35 to an oscillating state that can have a
71:37 pattern now that can have
71:38 a wavelength or it can be homogeneous
71:40 but it can be oscillating
71:43 and that’s one of the prime examples of
71:44 the type three instabilities
71:46 now the type two instability in the
71:48 middle is a little bit
71:49 uh subtle because here you have
71:52 the the value factor q so the
71:56 homogeneous state
71:57 is always marginally marginal uh
72:00 marginally unstable the others has this
72:03 sigma of q
72:05 uh of of zero so it doesn’t really know
72:08 whether to grow or shrink
72:10 and then as you increase your control
72:12 parameter
72:13 another wavelength becomes important
72:16 and ultimately dominates the system
72:20 if your value of r is large enough so
72:23 here you can have both so it’s not
72:24 really clear what you get
72:25 you can have a uniform pattern or you
72:28 can have something that is just a very
72:30 large pattern with a very large long
72:32 wavelength
72:34 so and what you see here these three
72:36 kinds of qualitative instabilities that
72:38 you get
72:39 of how you can get from a homogeneous
72:41 state
72:42 to a pattern state that is described by
72:45 the wavelength or by a wavelet away
72:47 vector
72:48 resembles some kind of universality
72:52 and why did we get here in rosalita why
72:54 is there why are there only these three
72:56 types where
72:57 can we understand a large class of
73:00 dynamical systems
73:02 by just three classes the reason is that
73:05 we restricted ourselves to situations
73:07 that look like this here where we
73:10 linearize
73:12 where we can linearize around this
73:15 homogeneous state
73:16 and then suddenly when we linearize all
73:19 other complexities
73:21 become unimportant yeah
73:25 so this type of instability that you get
73:27 tells you a lot about what kind of
73:29 pattern
73:30 you have

slide 15

73:33 now let’s have a look at a simple
73:35 example
73:37 yeah so i already mentioned briefly the
73:40 swift hohenberg equation
73:42 now that’s this one here
73:46 we have the second order derivatives and
73:48 the fourth order
73:50 derivatives and then we have a linear
73:53 term
73:54 in phi and a non-linear sorry
73:59 that’s fine
74:03 and the normally determined file and now
74:06 we make this ansas
74:08 that phi is equal to the homogeneous
74:10 state
74:12 plus some perturbation around this
74:16 and uh what we now do
74:21 what we now do is we linear-wise we say
74:24 that
74:25 we just look at very small perturbations
74:27 around this how much in the state
74:29 and we make our answers
74:37 we make our own dots at delta phi
74:41 is equal to some constant a either we
74:44 don’t know
74:45 e to the power of sigma q times t
74:49 e to the i q x
74:53 and now we substitute this unless
75:00 into this swift homework equation
75:09 and what we get is now a relation
75:12 between sigma q and q
75:16 ah so what we get is what is called
75:19 dispersion relation we got a relation
75:22 sigma q
75:25 r minus q squared minus 1
75:28 squared now that’s our dispersion
75:32 relation
75:33 and what you have in this dispersion
75:36 relation here
75:38 you can see on the left hand side that’s
75:40 that’s what you get if you plot it
75:42 now so for small values of r you have
75:45 this blue shape
75:46 and as you increase r you get
75:50 larger this this function moves up
75:53 that’s just a constitute moves up
75:56 and then at some point you pierce
75:57 through this point
76:00 at a certain value of qc
76:04 yeah so
76:07 at this value of qc
76:11 dominant wavelength
76:14 or wave vector
76:22 is just equal to 1.
76:26 now we ask what is the growth rate
76:31 at the maximum so the growth rate
76:39 at the maximum at the maximum
76:43 of the real part of qc
76:47 well and this is just of cube or sigma
76:50 sorry
76:50 sigma q
76:54 just plug that n we get r
77:00 yeah so this maximum moves linearly up
77:03 and of course you could have already
77:04 guessed that just from the
77:05 shape of this here
77:09 yeah so what this means is that we get
77:14 pattern formation
77:19 for r
77:23 larger than zero no for r larger than
77:27 zero
77:27 we have a dominant wavelength we have a
77:29 perturbation
77:31 and this perturbation in this linear
77:34 approximation
77:35 grows if r
77:39 is larger than zero yeah then we have a
77:42 periodic perturbation and also that of
77:45 some wavelength
77:46 we put a different kinds of parotid
77:48 deviations of perturbations with
77:50 different wavelengths like short
77:51 wavelength
77:52 long rate law wavelength and then we see
77:55 which
77:56 of these perturbations survives and
77:58 which has the fastest growth rate
78:00 and that’s what we say gives us the
78:02 pattern on the long
78:03 time and on the large scale now so here

slide 16

78:07 is a computer simulation of this
78:09 equation now
78:11 equation and this is the kind of pattern
78:14 that you get
78:15 and get to see in these equations and
78:18 of course i didn’t tell you anything
78:21 about
78:22 how this pattern looks like what i the
78:25 only thing
78:26 that this linear stability analysis
78:27 gives you
78:29 is that you get the wavelength now so
78:32 you got here
78:33 the uh typical length scale
78:36 of such a pattern and you get the
78:39 conditions
78:40 under which such a pattern can emerge
78:43 now and in our case as the toneberg
78:45 equation we get these kind of patterns
78:48 once r is larger than zero
78:51 and of course the systems are more
78:52 complicated there’s always a dynamic
78:54 belief that’s only for example you have
78:56 to make sure that you understand
78:58 what’s going on at the boundaries you
79:01 know so these boundaries can be very
79:03 very
79:03 important in selecting between
79:06 different kinds of patterns
79:11 okay so to conclude let me see okay
79:14 to conclude let me just have a look at
79:16 the time

slide 17

79:20 oh okay so we’re we’re talking quite a
79:22 while
79:23 okay so to conclude just give let’s give
79:26 me
79:26 let me give you another example here
79:28 without going to mathematical details
79:30 but it’s a very important example and
79:32 this example is called a reaction
79:34 diffusion equation
79:36 and it’s called the reaction diffusion
79:37 equation because it consists
79:39 describes systems that consist
79:42 of reactions and diffusion now so for
79:46 example
79:46 here the time evolution of our field
79:49 phi of x t is described
79:52 by some local function or some local
79:55 reactions
79:56 they are for example susceptible to
79:59 infected
79:59 also some local reactions
80:03 plus a diffusion term plus diffusion so
80:05 random motion
80:07 and so so if i told you that the
80:09 susceptible
80:10 this successful and affected people
80:13 were running around randomly in space
80:16 then
80:17 this dynamics would be described by a
80:20 direction
80:21 diffusion equation so these are the two
80:23 components of a reaction diffusion
80:24 equation
80:26 and these equations have a famous
80:29 result that is named after alan turing
80:33 and what he showed is that you need an
80:34 erection diffusion system
80:36 you can get patterns so that at this
80:38 point people
80:40 were did not believe that you have
80:42 diffusion
80:43 now you have diffusion something that
80:45 smooths down everything
80:47 and you can get a pattern and alan
80:51 turing
80:52 studied the conditions under which you
80:54 can get patterns
80:55 in such reaction diffusion systems and
80:58 what he basically said is that you need
80:59 at least two components there will be
81:01 two chemicals
81:03 and then he wrote down equations of this
81:07 form
81:09 and then he did exactly what we did now
81:12 for this general equation what we did in
81:13 the previous
81:14 minutes namely it conducted a linear
81:18 instability analysis so you linearize
81:21 these equations and if you linearize a
81:23 general equation
81:24 you get here derivatives or some
81:27 jacobians
81:28 now of these functions and you get the
81:31 conditions that relate
81:33 the jacobian so the the linear behavior
81:36 of these functions
81:37 with the diffusion constant yeah and
81:39 what he then said
81:41 is okay if you want to have a pattern in
81:44 the erection diffusion
81:45 system with two components then one
81:48 species needs to be an activator
81:51 so it needs to be positively regulating
81:53 itself
81:54 and the other species needs to be an
81:57 inhibitor so it’s negatively regulating
81:59 itself
82:00 and the other activator
82:03 and the second condition is that this
82:06 activator diffuses very
82:08 fuses very slowly and this inhibitor
82:11 diffuses fastly
82:15 so how can you get a pattern with that i
82:17 don’t go through the calculation here
82:19 but the way you get a pattern here is
82:21 that you have a homogeneous
82:24 system and you have a little
82:26 perturbation
82:27 on the wavelength like we did in the
82:28 mathematical enzymes
82:30 then this activator activates itself
82:34 yeah it will grow but it
82:37 this activation but it will not smear
82:39 out you know the diffusion of this
82:41 activator
82:42 here is low while the inhibitor
82:46 also gets activated but it diffuses away
82:50 so locally the activator can build up
82:53 the concentration peak while the
82:55 inhibitor
82:57 spreads out and that’s how you get a
82:59 pattern
83:00 in such a touring system and the
83:04 applicability of such turing systems is
83:06 of course
83:07 limited by this conditions here
83:10 now you need to have a diffusion cons
83:12 you have two components you need to have
83:13 a diffusion
83:14 difference of the diffusion constant of
83:17 a factor of 10 or so or 40
83:20 to see these effects and this is very
83:22 difficult
83:23 to achieve in biological systems
83:26 yeah one example where this seems to be
83:29 implemented
83:30 is lymph development i think that’s
83:31 chicken here
83:33 where you see i think that’s at the wing
83:34 of a chicken
83:36 how this evolves in
83:39 early development you know or in
83:41 development
83:42 of a chicken on the embryo and you can
83:45 see
83:46 here that you have these red regions
83:49 are regions where certain genes are
83:51 expressed now that are important for
83:53 development of bones or something like
83:55 this
83:56 and you can see how here this pattern
84:00 this touring pattern is established but
84:03 again like in the previous cases you
84:05 have a specific wavelength
84:08 yeah you see you know that gives you
84:12 a specific size of your body parts
84:15 of these fingers and
84:19 how does it work here despite having
84:23 this strong assumption on this
84:26 difference and diffusion constant
84:28 so this difference in diffusion
84:29 constants you only need if you have
84:31 really two components now if you have
84:33 four or ten components
84:35 then of course you can get an
84:37 instability
84:38 that gives rise to a pattern in a
84:41 touring system
84:42 even for um much weaker differences
84:46 in these diffusion content and in
84:47 biologically relevant contexts
84:51 okay so with this at uh i’d like to
84:53 finish so next week we’ll start
84:56 digging more into epidemics or using
84:58 epidemics as an excuse
85:01 to do some non-equilibrium physics and
85:04 i’ll hang around a little bit if your
85:06 case is there
85:07 there are any questions otherwise see
85:09 you next week bye 85:28 <br> [Music
85:36 oh there’s a lot of things going on in
85:37 the chat
85:40 um