introduction
00:00 to our little uh lecture
00:03 so meanwhile uh most of my group
00:07 are in current time and i'm one of the
00:09 few people left
00:11 and um and ethan detang who used to join
00:15 me here in the lecture hall
00:17 uh has escaped to a safer place
00:20 and so now uh buddy tang is watching the
00:24 chat and
00:25 the participantsness and we'll let you
00:26 in and
00:29 moderate a little bit the chat and um
00:34 so it's a good time actually to be
00:35 working on some epidemic models that we
00:37 start
00:38 that we started working on last time and
00:42 uh in particular last time
slide 1
00:46 we were looking at the simplest possible
00:49 epidemic model you might think about
00:52 so let me share the screen just to give
00:54 you a little bit of an
00:56 um
01:01 of reminder
01:10 okay
01:32 okay
01:39 okay here we go so we introduced this
01:43 little epidemic model where you had
01:46 two kinds of people infected one
01:50 and the susceptible ones and the model
01:53 was very simple so these infected ones
01:55 could uh meet up with an affected one uh
01:58 within with a susceptible one
02:00 that's that maybe they weren't uh to a
02:01 party or so it just
02:03 used the same train yeah and then the
02:05 susceptible one uh
02:07 got infected and then you have the
02:11 second process
02:12 where if you are infected with a certain
02:15 rate also probability
02:17 per unit of time you get rid of your
02:20 disease
02:21 and you recover and go back to
02:23 susceptible
02:25 you know so you're a recovered person
02:28 so and then we went one step further and
02:31 we
02:31 put this very simple model on a lattice
02:34 in a spatial context
02:36 the simplest spatial context you can
02:38 think about is just having
02:40 lattice in one dimension yeah
02:43 so suppose that this letter looks a
02:45 little like the train like a train
02:47 and uh so on this little letters you
02:50 react
02:51 basically with your nearest neighbors if
02:53 you if you infect somebody
02:55 you're in fact your nearest neighbor but
02:57 uh not
02:58 anybody else yeah and then
slide 2
03:02 we saw that such a simple model gave
03:04 rise
03:05 uh to rich phenomenology depending on
03:08 the relation between the rate of
03:10 infection
03:11 and the rate of recovery which was set
03:14 to one
03:15 now if the infection rate was very large
03:18 then
03:18 we got a state where we had a large
03:22 fraction of individuals constantly
03:24 carrying the disease
03:26 and when the rate of infection was very
03:29 low
03:30 then we ended up in a stainless
03:31 so-called absorbing state
03:33 where the disease got extinct
03:37 now in between that we found that there
03:39 is a value of this infection rate
03:41 where infection and recovery just
03:43 balance
03:44 and this state was characterized uh
03:47 very in a very similar way to the ising
03:50 model
03:51 in equilibrium was characterized by
03:54 scale and variance
03:55 that means that these correlation
03:57 lengths that we looked at
03:59 along the space axis
04:03 and along the time axis we had two
04:05 correlation lengths
04:06 they both diverged that means we had
04:09 self-similarity
04:10 in this critical point
slide 3
04:15 yeah and then we went on and we derived
04:17 the launch of the equation
04:19 and from this launch of an equation we
slide 4
04:21 then just wrote down
04:22 the martin citra rose functional
04:26 integral and i'll show you that later
04:28 when we actually used it
04:30 when we'll actually use it for the
04:31 realization procedure
04:33 and for once we once we had written down
04:36 these actions at the margin et cetera
04:38 rose function integral
04:40 of course we can go to free of space and
04:42 write down
04:43 this action also in fourier space
slide 5
04:47 now that's the formalities yeah
04:50 that looks pretty complicated now
04:55 we have a problem right so this looks
04:57 pretty complicated
04:59 and uh we have a situation
05:02 where we have divergences where we our
05:06 correlation length is infinite infinite
05:10 and we don't really know how to deal
05:12 with this infinite correlation
slide 6
05:16 and this is exactly the situation where
05:19 we have developed in the last
05:21 70 years or so mathematical techniques
05:25 originally in
05:26 quantum field theory that allow us to
05:29 deal with these
05:30 divergences with these infinities
05:33 now we've made two observations here so
05:36 one is
05:37 scale scale invariance that i just
05:38 mentioned yeah scale variance
05:40 at this critical point there's no
05:43 distinguished length scale we zoom
05:45 in and the picture we get is
05:48 statistically
05:49 still the same as the original picture
05:54 the second observation is
05:57 that so and as a result of the scale
05:59 invariance we saw that
06:00 in equilibrium all of these
06:03 thermodynamic quantities
06:05 i like the like the free energy
06:08 the magnetization and icing model obey
06:10 power laws
06:12 as you approach the critical point so
06:14 they diverge with certain power laws
06:17 and the second observation is
06:20 that at this critical point
06:23 our empirical observation is for example
06:26 in the equilibrium but also in
06:27 non-equilibrium
06:29 that there are a lot of different
06:32 real world systems that are described by
06:36 the same
06:38 theory the same simple theory so for
06:42 example
06:43 you have the icing more you know the
06:44 icing model is super simple it's much
06:46 simpler than actually a real magnet
06:49 nevertheless the icing motor can predict
06:52 exponents
06:54 near the critical point of real magnets
06:58 also of different materials very nicely
07:03 and it even the icing model even can
07:05 predict
07:07 critical phenomena in completely
07:09 unrelated systems
07:12 like phase transitions in water if you
07:14 put water in a high pressure then you
07:15 have a phase where you have
07:17 like a coexistence of water and steam
07:20 and this you can predict these exponents
07:22 you can predict with the icing water
07:24 and this power of such simple models to
07:28 predict a large
07:29 range of critical phenomena is called
07:33 universality and both of these
07:36 observations
07:37 are the scale and variance the cell
07:39 similarity
07:40 and universality can be systematically
07:43 started with a renormalization group
07:46 the first thing you might actually think
07:47 about okay so why do we need actually
07:49 something complicated
07:51 now we just do a naive approach
07:54 uh similar to the one that we used
07:57 implicitly in these lectures about
07:59 pattern formation
08:00 and non-linear dynamics now what we
08:03 could do is okay we say okay so we just
08:06 instantaneously zoom out of the system
08:09 yeah go to
08:10 wave vector zero so in finite wavelength
08:13 you look only at the very large
08:15 properties
08:16 and just pretend that we can average
08:18 over everything we just look at average
08:20 quantities
08:21 that's that should be called mean field
08:23 theory where you pretend
08:25 that your system the state of the system
08:28 at a certain
08:28 site is governed by
08:32 a field that rises over
08:35 basically an average over the entire
08:37 system
08:38 yeah so this mean field theory where you
08:41 instantaneously
08:42 go to the macroscopic scale
08:46 this doesn't work in others it fails to
08:49 predict these exponents that we get at a
08:51 critical point
08:52 and it also doesn't give any uh reason
08:55 why we should have universality
08:59 now renewalization does something very
09:02 smart now that provides a systematic way
09:06 of connecting these microscopic degrees
09:09 of freedom that are described for
09:10 example by hamiltonian
09:13 to uh macroscopic descriptions
09:17 now it does so by taking into account
09:21 all the intermediary scales between
09:24 micro
09:24 the microscopic level and the
09:26 macroscopic level
09:28 and as you uh my guess from these
09:30 critical phenomena if you have scale
09:32 environments and all
09:33 scales are equal and equally important
09:36 this approach where you
09:38 take into account all of these scales
09:41 one at a time
09:43 now going from microscopic to
09:44 macroscopic which
09:46 makes much more sense than arbitrarily
09:48 focusing only on the largest scale as
09:50 you do in new york theory
09:54 now so rememberization group the
09:56 romanization
09:57 group allows us to go from the
09:59 microscopic
10:00 level now that is described by some
10:03 hamiltonian
10:04 by some symmetries all the way to the
10:07 macroscopic
10:08 level without ever forgetting what is
10:11 going on
10:12 in between now that's the power of the
10:14 renalization group and in this lecture i
10:16 will show you
10:18 how this is actually implemented
10:21 now how we can actually do that
10:30 okay so there's a little bit of a lot of
10:33 information i should have split this
10:34 slide
slide 7
10:35 um so
10:39 so so we are now going to follow this
10:41 program now going from microscopic
10:43 to macroscopic one scale
10:46 one length scale at a time now we're
10:49 starting with something microscopic that
10:51 is super complicated don't think about
10:53 the icing model
10:54 think about something that has 10 000
10:57 different couplings or so
11:00 now we take that model with 10 000
11:03 different parameters
11:05 on the microscopic scale so the real
11:07 physics
11:08 that's going on on the microscopic scale
11:10 that's complicated
11:12 yeah it's much more complicated than the
11:13 icing model and it just as the disease
11:16 is much more complicated as the ice i
11:18 model that i showed you
11:21 and then we go to larger and larger
11:24 scales
11:26 and hope to end up with something that
11:28 is simpler
11:30 and less correlated than on the
11:32 microscopic scale
11:34 yeah so how do we do that so realization
11:37 consists of three
11:38 steps uh the most important step
11:42 now the the actual uh what's actually
11:46 underlying renovation
11:48 is a course grading step now in this
11:51 course grading step
11:53 you can think about uh that you
11:57 unsharpen an image suppose you take a
11:59 photo
12:00 and then you can focus you can make the
12:02 picture
12:04 uh sharp or less sharp yeah by turning
12:07 like the lens
12:08 and the ring on the lens if you still
12:10 use a rear camera
12:11 yeah so you can make a job or let's try
12:13 or think about the microscope
12:15 you can have to turn some knobs to make
12:17 the image sharp or not sharp
12:19 and the first step we
12:22 cause grain so we cause grain
12:26 and that means literally that we make
12:28 the image that we're looking at in the
12:30 system
12:31 unsharp so that we that's what we do
12:35 yeah so we
12:36 mathematically that means if you uh if
12:39 you close grain if you make something
12:41 out sharp unsharp then that means that
12:43 you integrate out
12:46 fast fluctuations or short range
12:48 fluctuations
12:49 i'll show you on the next slide how this
12:51 looks like intuitively so first you get
12:53 rid of
12:53 all these uh fluctuations that happen on
12:56 very small length scales
12:59 and you have to perform an integral to
13:01 do that
13:02 and this integral of course is very
13:05 complicated
13:06 to calculate now the second state
13:10 we have a new field yeah
13:13 now let's say phi average
13:17 but because we have course grade this
13:19 new field
13:21 our new image is blurry but
13:25 it is also not on the same length scale
13:27 as before
13:29 yeah so it's just blurry but what we now
13:32 have to do
13:33 is to rescale length
13:36 to make the structures that we have in
13:38 this new image
13:40 similar to the structures that we had in
13:42 the original image
13:43 now so that means we need to rescale
13:45 length scales
13:48 by a factor of b
13:52 and the other thing that happens if you
13:53 make things blurry
13:55 is that you lose contrast in the image
13:58 so the image looks a bit dull so we
14:02 also need to now to increase the
14:03 contrast again
14:05 and we do that by rescaling our
14:09 fields as well so these are the three
14:12 steps of renormalization as if we first
14:14 integrate out very short
14:18 fluctuations happening on the smallest
14:20 length scales
14:22 and the second step and the third step
14:25 make sure that once we have integrated
14:28 out
14:29 these uh these short length skills
14:32 that our new theory that we get is
14:34 actually comparable
14:36 to the previous one so we have to reset
14:39 the length scales and we have to reset
14:41 the fields the contrast of the field by
14:44 multiplying them
14:45 with appropriate numbers
slide 8
14:50 so how does that look like so we start
14:52 with a field file
14:53 now that has short range fluctuations
14:56 you know fast fluctuations in space
14:58 now for example um
15:02 yeah here these wobbly things yeah these
15:04 are fast fluctuations
15:07 but it also has slow fluctuations that
15:09 would be
15:11 that would be here a slow fluctuation
15:15 now you have fluctuations on all
15:17 different length scales and now
15:19 we cause grain that means we integrate
15:21 out
15:22 these fast fluctuations and what we get
15:25 is a new field if i uh this phi
15:29 average here because phi r
15:33 that is smooth that doesn't have these
15:35 small
15:36 wiggles anymore but it is smooth here
15:41 now so we have course grade the fields
15:44 and now we have coarse grains the field
15:46 we reset the length
15:48 rescale the length scale
15:51 x prime to make fluctuations on the new
15:57 field comparable to typical fluctuations
16:00 on the
16:00 original field the second step
16:05 we then renormalize um
16:08 there's actually no three here because i
16:11 started
16:12 i know there is a three okay so uh
16:16 as a final step and we have to rescale
16:18 the fields now we have to
16:20 change the magnitude of the signal here
16:23 to make it comparable
16:24 to the original signal over here
16:28 now this sounds very intuitive
16:31 and very simple but of course in reality
16:34 it's quite difficult
answering a question
16:39 so let's see how this works uh
16:42 what's the result so can i have a
16:45 question here
16:50 um in the previous slide
16:54 when we re-normalize uh fi
16:57 prime we essentially doing it
17:01 because we want uh whatever abs
17:04 magnitude or value of phi prime is to
17:07 equal
17:08 uh the value of five the original field
17:11 yes
17:11 exactly so what we want to do is so so
17:14 we do this procedure not only once
17:16 but many times now the next step will do
17:18 that many times
17:20 and uh each time we do these three steps
17:24 we get a new hamiltonian or a new action
17:28 and this new action will be
17:31 different to the original action
17:35 it will be different for trivial reasons
17:38 namely because once we cause when
17:41 think about this course grading as an
17:43 average instead
17:44 now you have i'll show you later a
17:47 specific example suppose you average
17:49 uh in some area here and that's why
17:52 that's how you smooth
17:54 yeah when you average you know think
17:56 about the spin system you average
17:59 then you don't have plus minus plus one
18:01 or minus one
18:02 but the new average field will be plus
18:04 0.1
18:06 and minus 0.1 now just because you
18:08 averaged
18:09 over in the field yeah but you don't
18:11 want the new spins to live in this world
18:13 of plus
18:14 0.1 and minus 0.1 but you'd want them
18:18 also
18:18 live on the scale of plus minus 1 just
18:21 to make the
18:21 hamiltonians comparable now so this is a
18:24 trivial effect that you get by course
18:26 grading that you don't want to have
18:28 to dominate your results and we need to
18:31 get rid of that
18:32 this trivial rescalings of the fields
18:35 and of the
18:36 of the um of the length scales just by
18:39 explicitly taking the step and saying
18:41 okay
18:42 now i have co-strained my field now i
18:45 have to reset the length scales
18:46 and i have to reset the amplitude of my
18:49 fields
18:50 by multiplying these these quantities
18:53 with appropriate numbers
18:56 i just want to have comparable things at
18:58 each step
19:00 yes thank you so much uh but regarding
19:03 this particular point
19:04 you showed in the first slide that phi
19:07 prime
19:07 phi is divided by b uh in the
19:10 in introductory slide remember
19:12 normalization i couldn't understand that
19:15 yes here yeah five in the next one
19:19 five bar by b or um
19:23 so which which one third point
19:26 third point yeah that's just a number
19:28 yeah we don't know this number yet
19:31 we don't know it yet we can get it by
19:33 dimensional analysis
19:34 uh very often um so we don't know this
19:37 number yet
19:38 it doesn't the same b okay it's not
19:42 from one i think it's an s actually
19:47 so so it doesn't have to be b it depends
19:50 on the dimensions of your field
19:53 typically b to the power of something
19:55 also
19:56 that doesn't have to be b and it depends
19:58 on the on the dimensions of your fields
20:00 here
20:01 um okay just at this point we just say
20:03 okay so we have to do something with our
20:05 fields to make them comparable
20:08 now think about an average now you can't
20:10 get average and if you're doing average
20:12 all the time
20:13 yeah then your uh the central limited
20:16 theorem will be that the
20:17 average like in a disordered system will
20:20 get very very small
20:22 yes the variance of this average will
20:24 get very very small
20:26 and we don't want this effect to happen
20:28 because it destroys this basically
20:30 or if i what we actually want to look at
20:33 now what we actually want to look at is
20:34 how does the theory in itself the
20:36 structure of the theory
20:38 change as we go through the scale and we
20:41 don't want to have these effects that
20:42 come
20:43 by uh by uh
20:46 just that we that we can't compare
20:49 um suppose you compare like uh uh
20:53 you compare um the velocity of a car
20:56 you live in the uk also or in the united
20:57 states and you compare the velocity of
20:59 the car in miles per hour
21:01 or in kilometers per hour now you have
21:04 to you cannot compare that
21:05 but pure numbers you have to do
21:07 something you have to be scared by 1.6
21:09 to make them comparable and here also we
21:12 go through these scales look
21:13 like meters uh kilo kilometers miles and
21:17 so on
21:18 and to make these things comparable all
21:20 the different lengths that we have to
21:21 rescale
21:22 them all the time yeah just that way
21:24 that we're all the way talking about
21:26 um the same thing um
21:30 so in germany we say we uh you cannot
21:32 compare apples and oranges
21:34 different things you have to uh
21:37 if you compare apple and origins then um
21:40 then then you're doing something wrong
21:43 in other words to compare apples
21:45 to different kinds of apples so to say
21:48 yes but we always want to talk about
21:50 apples and not about miles and
21:52 kilometers now so that's that's the idea
21:55 about this rescaling step
21:59 i'll show you later an example where
22:00 this rescaling step is already implicit
22:02 of course you can choose this course
22:06 grading step in a way
22:07 that it doesn't change the magnitude of
22:10 the field
22:12 yeah so that you can choose the course
22:14 grading step in a way here this one
22:16 in a way that it doesn't change the
22:18 magnitude of this field and then you
22:19 don't have to do this renormalization
22:21 step
22:24 now but the principle you will have to
22:26 do that
22:28 okay so now we have these three points
22:31 now we rescale uh we of course grain
slide 9
22:35 rescale and renormalize and once we do
22:39 that
22:39 our action or our equilibrium or
22:41 hamiltonian
22:43 will become a new action
22:48 now s prime and say we
22:52 did this rescaling on a very small
22:54 length scale
22:55 dl now this s prime
23:00 is then given by some operator r
23:03 of s and if we do that repeatedly
23:08 you know so what we then get is a
23:11 remoralization group flow our g
23:13 flow and that's basically
23:17 the differential equation for the action
23:21 yeah the s over the l
23:26 is then something like this r
23:29 of s so we normalize one step further
23:33 minus the previous step
23:38 you know so we change we do these three
23:41 steps
23:41 just a little bit not just and we call
23:44 square
23:45 we integrate out a very small additional
23:48 scale
23:50 yeah and our action is then different on
23:53 the next scale
23:54 of course we assume that this is somehow
23:55 continuous and well behaved
23:58 and then we'll have a flow equation
24:01 of our action and of course in reality
24:04 this will not be a flow equation of our
24:06 action
24:06 but of the parameters that define our
24:09 action
24:10 now suppose so how does it look like
24:18 now so here is in some space of all
24:21 possible actions
24:23 of some parameters p1
24:27 p2 p3
24:32 now and this action now think about the
24:34 uh
24:35 think about our non-linear dynamics or
24:37 dynamical systems lecture
24:39 what we can this is this is a
24:41 differential equation here
24:44 it looks complicated but this is a
24:46 differential equation it tells you
24:47 how the parameters of our action
24:51 change as we cause gradients we do this
24:54 renormalization procedure
24:57 now and this gives us a differential
24:59 equation
25:01 and if you have a differential equation
25:02 that will be highly non-linear of course
25:05 now we can do the same thing as we did
25:07 in this lecture or nonlinear system we
25:09 can
25:10 derive a phase portrait now in the space
25:12 of all possible actions
25:14 of all possible models where does this
25:17 minimalization group
25:18 group flow carry us
25:22 so let's start with a very simple
25:27 line
25:31 that's the line in the space of all
25:33 possible models
25:34 now this is the line of models
25:37 that actually describe our physical
25:39 systems
25:40 now think about different combinations
25:42 of temperature
25:43 and magnetic fields in the icing model
25:48 now so this is this is where this model
25:50 lives in this space
25:52 now if we take the right parameter
25:55 combinations
25:58 we will be at some critical point
26:06 yeah so there's no flow yet now so this
26:09 is just
26:10 uh the the the the range of different
26:13 models that we can have for example in
26:15 eisenhower that correspond to some real
26:17 physical system
26:18 but there of course there are many other
26:20 model in the models in this space
26:22 that don't describe our physical system
26:24 now they don't describe a magnet but
26:26 something else or that there are not
26:28 given by a simple hamiltonian with
26:29 nearest neighbors
26:30 interactions but by something that has
26:33 long-range interactions or something
26:35 very weird
26:36 now there's a space of all possible
26:37 models is very large
26:40 now we have this critical point and
26:43 other models in this space also have
26:46 critical points
26:47 now and these critical points live on a
26:51 manifold
26:58 now they leave on a manifold you know
27:01 that is the critical manifold
27:09 you know so that's all the points of
27:11 this critical manifolds are critical
27:13 points
27:14 of some actions
27:17 and our action our critical point of our
27:20 action
27:21 is also on this manifold but all
27:24 other points of this manifolds are also
27:26 some critical points of some other
27:29 actions so
27:33 what happens now now
27:37 we are close to the critical point let's
27:39 say
27:40 we're here now very close to the
27:44 critical point
27:46 and now we really normalize
27:50 we go through we make this procedure of
27:53 renormalizing
27:54 the core straining going to larger and
27:56 larger scales and rescaling our fields
27:59 and lengths
28:00 and then our anatomy or our action will
28:03 change
28:04 so it will flow in this space
28:07 in some direction
28:11 so where does it flow to in the
28:13 non-dynamics lecture
28:16 we've seen that what determines such
28:18 dynamical systems
28:20 are fixed points and
28:24 what you typically have to assume in the
28:25 real normalization group theory
28:28 that there is some fixed point
28:31 on this critical manifold yeah
28:36 the fixed point of the realization would
28:39 flow
28:40 on this critical manifold
28:44 and now what happens with our flow
28:50 of course we will go to this fixed point
28:55 and then we might go away again
28:59 now the sixth point the fixed points
29:02 like in the dynamic resistance lectures
29:04 determine the flow of our dynamical
29:07 system
29:09 now what is this fixed point here this
29:11 fixed point is not the critical point
29:14 it's the critical point of some other
29:16 model
29:18 but this critical point here
29:23 has stability yeah just like in normal
29:25 domains so this is non-linear dynamics
29:27 here
29:28 so it's very often exactly what we did
29:30 in this lectures before
29:31 so we asked what is the flow now we ask
29:33 them to ask about the stability
29:35 of this fixed point now so this fixed
29:39 point
29:40 has stable directions now you perturb
29:44 and you get pushed out and unstable
29:46 directions
29:48 yeah typically or by definition
29:52 the directions on the critical manifolds
29:55 are stable now that's how this manifold
29:59 is actually defined
30:00 you can and there's a theory and only
30:02 resistance
30:04 that tells you that and there are also
30:07 other directions
30:09 that are not stable now let's have them
30:12 in green
30:14 for example the way i drew this these
30:17 are the ones
30:19 in this direction
30:26 so this determines the stability of the
30:28 flow of our fix
30:29 of our our system and if we have only
30:32 one fixed point this one fixed point
30:35 will tell us what happens to the flow of
30:37 our system just like in nonlinear
30:39 dynamics
30:40 lecture okay so
30:43 now we re-normalize
30:46 it because we didn't start exactly at
30:48 the critical point we stay in the
30:50 vicinity of this manifold here
30:52 here and this critic this fixed point of
30:55 the flow will do
30:56 something to us no it will push us away
30:59 or will attract us and now you can
31:04 do the same thing that you do in
31:05 learning your dynamics namely you linear
31:07 wise
31:08 around this fixed point so we say that
31:11 our action
31:13 [Music]
31:16 called linear stability
31:21 you linearize around this fixed point so
31:24 you say
31:25 that our action is equal to the action
31:29 at this fixed point
31:32 plus sum over different
31:35 directions i who operate is i
31:39 h i
31:45 times b to the power of lambda i
31:52 times
31:56 q i yeah so this here are
32:01 these two eyes are the eigen directions
32:08 so these are operators you can think
32:09 about this as operators so like eigen
32:12 direction of these operators and
32:16 the b is our course grading scale
32:30 the h tells us
32:34 how far we are away from the fixed point
32:42 and the s tells us uh is just the action
32:48 that we have at this fixed point
32:51 you know so we it's the same thing for
32:53 all of this was once we have once we
32:55 have this mineralization group flow
32:58 we're in the subject of non-linear
32:59 dynamics and we use the tools
33:02 of nonlinear dynamics renalization group
33:05 theory this language is slightly
33:09 different
33:10 you know so that's that's why you have
33:12 this b to the lambdas and so on
33:14 here and you separate the h i from the b
33:17 to the lambda
33:18 that's just the framework the how you
33:20 write it in the realization
33:23 theory for convenience reasons but what
33:26 we do here is
33:27 a simple linear stability analysis
33:30 of a non-linear system i will
33:34 re-linearize around the fixed point
33:36 we see how whether perturbations grow or
33:39 shrink
33:40 in different directions yeah and that
33:43 characterizes
33:44 then our nonlinear system and now we can
33:47 ask if we perturb around this fixed
33:50 point
33:51 in one direction i does it grow
33:54 this perturbation or does it shrink
33:57 lambda i
33:58 is larger than zero now this bi
34:03 is larger than one or b is larger than
34:07 one
34:08 now so lambda i is larger than zero then
34:11 our perturbation will grow
34:13 yeah perturbation
34:20 growth and then we say this direction or
34:23 this operator q
34:24 i you can also think about q i as one of
34:27 the
34:28 terms in the action now think about one
34:31 of the terms in the action
34:32 or one of the terms in the hamiltonian
34:36 this direction qi
34:40 is then called relevant
34:45 why is this relevant what we call when
34:46 do we call this relevance
34:48 you know if this perturbation grows we
34:51 are in this green direction here that
34:53 pushes us away from the critical point
34:56 so that means
34:56 that if we are an experimentalist
35:00 and if we want to tune our system
35:03 to get into the critical point
35:07 then
35:10 then we know that we have to turn
35:13 these relevant parameters qi
35:18 now this relevant parameter for which
35:19 this uh that are
35:21 unstable directions of this fixed point
35:25 now and if we have a relevant directions
35:28 we also have irrelevant directions
35:30 lambda is smaller than zero
35:32 and these directions are called
35:34 irrelevant
35:35 now perturbation
35:43 shrinks that means that
35:47 qi is irrelevant
35:53 that means the qi in this case is not a
35:56 parameter that drives us into this
35:58 critical point
36:01 and then we have the case that lambda i
36:03 is exactly equal to zero
36:05 then we don't really know what to do
36:08 then this q
36:09 i is
36:12 marginal and we cannot tell from this
36:15 linear stability analysis alone
36:17 from the linear realization around this
36:19 fixed bond we cannot tell alone
36:21 whether this perturbation will grow or
36:24 shrink and we have to use
36:25 other methods okay
36:28 so what happened now so we
36:31 started our theory close to the critical
36:34 point
36:35 we did this rememberization group
36:37 procedure
36:39 course grading rescaling renormalization
36:43 and then in this procedure
36:46 our action or our model will flow
36:49 through the space of all possible models
36:52 yeah
36:55 and then we ask where does it flow to
37:00 and we look at the non-linear dynamics
37:01 lecture and ask
37:03 so where does such a nonlinear system
37:05 drive us to
37:07 and our nonlinear dynamics lecture will
37:09 tell us look at the fixed points
37:12 right and in this realization flow you
37:14 also have fixed points you assume that
37:15 you have a fixed point
37:17 and this fixed point tells us about
37:21 what is going on on the macroscopic
37:24 scale now that determines this fixed
37:26 point
37:26 determines which is the end result of
37:28 our minimization group
37:30 procedure
37:33 so at this fixed point is characterized
37:36 by stability
37:38 it has a finite number of relevant
37:40 directions
37:43 now it's had a finite number of
37:44 parameters that are actually important
37:47 to change if you want to go to the
37:48 critical point
37:51 and because you only have a finite
37:53 number of directions that are relevant
37:56 you usually get away with models that
37:58 also have this very finite number
38:01 of parameters instead of models like a
38:04 bigger magnet or so that is a very
38:06 complicated geometry and everything
38:08 there are instead of model that has 15
38:10 000 parameters
38:12 now you get away with a finite number of
38:13 parameters that are given
38:15 by the relevant eigen directions
38:19 of this fixed point now so now what
38:22 happens if we have a different model so
38:24 now this is now the
38:25 magnet one that this is not a
38:29 magnet or material one we can also look
38:31 at a magnet at another of another
38:33 material
38:34 now so this magnet one
38:40 and now we have here
38:47 a magnitude
38:51 now each of them at the physical real
38:53 microscopic level is destroyed by 15 000
38:56 parameters or whatever something very
38:57 complicated
38:59 and for this magnitude we can do the
39:02 same procedure
39:04 we renormalize
39:07 and we while we renormalize we will end
39:10 up at the vicinity of this fixed point
39:14 and in the vicinity of the fixed point
39:17 the behavior of the flow is determined
39:20 by a finite number
39:22 of parameters again
39:26 and both of these magnets here are
39:29 under renormalization now on the large
39:31 scale repeat these
39:33 procedures determined by the same fixed
39:37 point
39:37 about the same point but this one here
39:41 and because they're determined by the
39:42 same fixed point with the same stability
39:46 and with the same properties of how they
39:48 go
39:49 uh of how the flow behaves around this
39:51 fixed point
39:53 that's why these two magnets here are
39:55 described macroscopically by the same
39:57 theory
39:58 and that's then the reason why we have
40:01 universality
40:03 so in this way in this very general way
40:05 so we look of course in our in
40:07 more detail the renal isolation group
40:10 theory gives us a justification for why
40:14 only a finite number of parameters
40:17 matter
40:17 on the or finite a limited level of
40:21 description is sufficient to describe
40:23 large scale properties of a large
40:26 number of very different systems and the
40:29 reason is
40:30 at this critical point they're
40:33 described macroscopically by the same
40:36 fixed point of the renewalization
40:39 workflow
slide 10
40:42 now how does this look in detail
40:47 so the very first or the
40:50 most uh simple way of doing
40:53 remoralization
40:55 is to take what i said initially about
40:58 this
40:59 defocusing about this course training
41:01 literally
41:02 and do the whole procedure in real space
41:06 now suppose you have here a lattice
41:09 system
41:11 and also suppose you have this lattice
41:12 system here and you have some
41:15 spins here and what you can do then to
41:19 cause grain
41:20 is to
41:24 you know what you can do to coarse grain
41:26 is to create
41:27 boxes or blocks now of a certain size
41:31 and then to calculate this cross here
41:34 that is a representation of all of these
41:37 microscopic spin for each block
41:40 yeah so we have this uh spins here
41:44 like what are 16 in each block and you
41:48 now transform them to a single number
41:50 you can do that by averaging over them
41:52 or you can take
41:54 you can say that i take the spin
41:58 that is the majority of these other
42:01 spins here
42:02 if the majority goes up then my new spin
42:06 that describes the entire block
42:07 will also go up and this would be a way
42:11 to get a rid of this renormalization
42:14 step if you say i take the majority
42:18 i take a majority rule so this new spin
42:20 here x
42:21 will take the value of the majority of
42:24 the original spins
42:27 then the new spin will also be plus or
42:29 minus one
42:31 if my new spin yeah and i don't have to
42:33 renormalize them because it has the same
42:35 values as the original splits
42:38 if i take the new spin as the average
42:43 over all of these spins here then
42:47 i typically get a very small number the
42:49 average won't be one or minus one but it
42:51 will be
42:52 0.3 or 0.1 or 0.5 or so but it will not
42:57 be
42:57 one or minus one in most cases yeah so
43:00 in this case if i perform this procedure
43:02 i would have to renormalize
43:04 you know and i have to would have to
43:05 rescale my fields
43:08 to make them comparable to the original
43:10 step
43:12 yeah so now we have these blocks
43:17 and we define some new spin that
43:20 describes each of these blocks
43:23 and now we write down a new model
43:26 a new hamiltonian for these new spins
43:29 here
43:31 and what we hope is that the spins that
43:35 we have here
43:38 in this new system this course-grade
43:40 system are described by a theory
43:43 that is structurally very similar to the
43:45 original theory
43:48 and this hope is actually justified by
43:52 the observation of
43:55 scale and variance now so if your system
43:58 is scaled in variance we can hope that
43:59 if we zoom out
44:01 and our system is statistically the same
44:04 then then our partition function
44:06 or our action will also be the same
44:09 just with some different parameters now
44:11 that is the hope that is underlying
44:13 renewalization group procedures with
44:16 these
44:16 block spins here what you typically get
44:19 is that you get
44:20 higher order turns all the time you know
44:23 so that's this hope is not
44:24 mathematically super precise uh but
44:28 that's what you have to assume in order
44:30 to achieve anything
44:33 okay yeah
44:36 okay so we call screen and then
44:40 we rescale the second step so that the
44:42 distance between these
44:44 spins here the new spins is the same
44:47 distance as we had
44:49 between the original spins now that's
44:52 what we have to do anyway
44:53 and that's why how we divide length
44:55 scales
44:57 by the same factor that corresponds to
45:00 the size of our boxes
45:02 yeah and now the lengths are the same as
45:04 before
slide 11
45:09 so let's do this procedure in a very
45:13 simple case which is the 1d
45:16 ising model
45:21 now so the one deising model now is
45:24 written the tradition function
45:26 it can be written in like a long form
45:30 in this way here that i sum up
45:33 all combinations of nearest neighbor
45:36 interactions
45:38 now that's just the hamiltonian here of
45:40 the ising model without an external
45:42 field
45:43 now and then i have to have a sum about
45:45 all possible values of the sigma i's
45:48 of the of the that my fee that my all
45:51 the possible values
45:52 that the sigma can take that gives me my
45:54 partition function
45:57 now what i do know is the first
46:01 course grading stuff now this first
46:04 coast grading step
46:06 means that these red spins here
46:09 like with the all these these black
46:11 spins are the white splits
46:13 now i'll integrate out the white spins
46:17 here in this picture i'll integrate
46:20 these ones out
46:23 now every second spin all even spins
46:26 and if i do that i will get a new theory
46:30 that is described by interactions
46:32 between these
46:33 uneven spins that are and these
46:36 interactions occurred to us are
46:37 depicted in these red with these red
46:41 lines
46:44 okay so and the stars is actually very
46:47 simple activated talks
46:49 so that's so for if we just do it for
46:56 for sigma 2 now we just sum
46:59 out we take the terms that correspond to
47:02 sigma 2
47:04 and we get that that is equal to
47:08 not many terms and then we have the
47:11 contribution from sigma 2
47:14 uh k sigma 1
47:17 plus sigma 3 now that's what what's left
47:20 plus e to the minus k sigma 1
47:25 sigma 3
47:28 and then all the rest e to the k
47:32 sigma 3 sigma
47:36 4 plus four sigma
47:40 five and all the other splits
47:44 so what i just that said is that sigma
47:46 two can have two values
47:48 minus one plus one yeah and i just
47:51 substituted i expected
47:53 explicitly now set sigma 2 to plus 1 and
47:56 minus 1
47:57 and perform the sum and that's what i
47:59 get then here for this first
48:01 term and now i can do that for all
48:12 even sigmas and then what i get is
48:16 exactly the same thing sum over
48:20 many terms e to the k
48:23 sigma 1 plus sigma 3 as before
48:27 plus e to the minus k sigma 1
48:31 sigma 3. now that was the original one
48:34 where we set sigma
48:36 2 to -1 and then we get the same thing
48:42 e to the k for the next
48:45 term now for the next interaction sigma
48:48 3
48:49 plus sigma 5
48:54 e to the minus k sigma 3
48:58 sigma 5
49:01 and so on yeah
49:07 for all the other terms yeah
slide 12
49:13 so now the idea is
49:18 that because when we are at a critical
49:21 state
49:23 that we expect our partition function to
49:25 be self-similar when we call screen
49:28 the statistics of the system remains the
49:31 same as we zoom out
49:33 and that's why we also expect
49:36 the quantity that gives us the statistic
49:40 statistics the partition function to be
49:42 self-similar as well
49:45 and what we now do is that we find
49:49 a new value of k prime
49:53 and some function f
49:56 of k that tells us that we
49:59 that these terms that we got
50:02 here sigma 1 sigma 3 we always got the
50:06 sum for each coupling
50:08 they should take the same form as the
50:11 original hamiltonian but with some
50:14 pre-factor here
50:17 and some new coupling here
50:20 but the form should be the same as
50:22 before
50:24 now as i said this is not usually
50:27 well justified but we have to do that in
50:29 order to do anything
50:31 and if you require that if you do some
50:33 algebra you will find that if you set
50:36 k prime to this one one half
50:39 logarithm hyperbolic cosine of 2k
50:44 and the function f of k to this year
50:48 then it fulfills this condition yeah
50:54 so now we can plug this in now so if we
50:59 if we use that
51:01 then our hammer tilt our partition
51:05 function
51:06 will read again we have many
51:10 terms f of k
51:14 e to the k prime
51:17 sigma 1 sigma 3
51:21 f of k e to the k
51:24 prime sigma 3 sigma 5
51:29 and so on
51:32 now and this is just the same we can
51:35 write this now
51:36 as a new partition function that has a
51:40 new prefactor
51:42 f of k now we pull this out this
51:45 prefactor
51:46 f of k and we have that n over two times
51:51 times a new partition function that
51:55 depends on the new system size
52:00 and a new coupling k prime
52:04 yeah so we have this down this one
52:06 renormalization step
52:08 we get a new partition function that
52:10 looks exactly the same as the old one
52:12 in structure but we have a new coupling
52:15 k prime
52:17 and a pre-factor here that is this
52:20 function
52:21 f and that also depends on the coupling
52:25 also what we did here is that we now
52:27 have
52:31 a relationship between the partition
52:34 functions
52:35 at different stages of the
52:38 renewalization procedure
52:45 yeah and now
52:50 what does it mean look at
52:53 this one here k prime
52:57 this is already a description
53:02 now this is already a description of how
53:04 our coupling
53:05 one parameter k prime
53:09 depends on the value of this okay now
53:12 how this parameter k
53:13 evolves in this course grading procedure
53:17 in this renewalization procedure
53:20 you know so this k that gets updated
53:23 now it's not in differential form yeah
53:26 like in
53:26 like uh like we did in the non-linear
53:28 dynamics
53:29 actually but it's in this uh other way
53:32 that you can describe non-linear
53:34 systems but by iterative updating now so
53:38 the new value of k
53:39 prime is given by the old value
53:43 is by this function here applied to the
53:45 old value
53:47 and now now is this is this updating
53:50 scheme here
53:52 and we can expand this term here the
53:54 logarithm
53:55 of the hyperbolic cosine and so for low
53:58 values of this coupling
54:00 this goes with k squared now so normally
54:03 i have an idea
54:05 about how this looks like i have already
54:08 prepared this
54:09 very nice and we can now solve this
54:11 equation here
54:13 graphically so we want to get the flow
54:15 and we can solve this graphically
54:17 and see where this linearization
54:19 procedure
54:20 carries our k our coupling k
54:23 now and because our partition function
54:27 remains invariant
slide 13
54:30 well our k the update of our k
54:35 describes actually the behavior of our
54:37 hamiltonian
54:38 under renormalization
54:41 okay so this is the plot here so what i
54:44 what you do
54:44 is that you plot the left hand side
54:48 of this equation k prime it's just
54:51 linear with slope one
54:54 and uh the right hand side
54:58 now this is uh this here
55:02 and where the left hand side is equal to
55:04 the right hand side there you have a
55:06 fixed point
55:07 now so this is here the way you need to
55:10 read this this is the next value of the
55:11 realization
55:12 this is the previous one if you start
55:14 here we'll go
55:15 here then here and here and here
55:19 and at some point these two lines meet
55:22 and that's
55:22 that's where your fixed point is so this
55:25 humanization procedure
55:27 will bring us to some fixed point which
55:29 happens
55:30 to be uh down here at zero
55:36 yeah and then we can look at how this
55:39 uh we can also then plot
55:42 a flow diagram as i did in a more
55:45 complicated way before
55:47 and how we also did it in the actual
55:49 nonlinear dynamics lecture
55:51 now we can plot it on this in this
55:53 one-dimensional line
55:55 then we have a stable fixed point at
55:57 zero
55:58 now that's here stable
56:02 and any value where we start with our
56:04 course grading procedure
56:06 will be driven to a value of k equals
56:09 zero
56:10 now we start with a very strong coupling
56:12 we renormalize
56:14 and we will be driven to zero
56:17 that means that this renormalization
56:21 procedure this course grading procedure
56:24 in this one-dimensional icing model
56:28 will always on the macroscopic skin on
56:31 large
56:31 length scales always lead to a model
56:35 that is effectively described by
56:40 a system that has zero coupling
56:43 yeah this coupling here vanishes and if
56:46 we have zero coupling that means that
56:48 our system
56:49 is above the critical point that's
56:52 non-critical
56:54 so we will normalize we go on and on
56:57 and we always end up on the system that
56:59 has very high temperature
57:01 or very low coupling yeah that's a it is
57:04 a disordered system
57:06 and that's just a reflection of the fact
57:07 that the one deicing model
57:10 doesn't have any order for a finite
57:12 temperature
57:15 now so you have to start with coupling
57:18 exactly equal to infinity
57:21 to get an order over the temperature
57:24 exactly
57:24 equal to zero only then you can have
57:26 order everything else
57:28 will drive you to this fixed point here
57:32 that corresponds to a system where you
57:35 have no
57:36 coupling at all now so the one that we
57:39 knew that already that the 1d
57:40 system doesn't show all in the ising
57:42 model doesn't have order
57:44 it doesn't have a really critical point
57:47 and that's why our flow
57:48 tells us that macroscopic scales this
57:51 system
57:51 goes to a system that doesn't have
57:55 any interaction so it's completely
57:56 disordered
slide 14
57:59 of course you can do the same procedure
58:01 for um
58:04 for the 2d system yeah and there is of
58:06 course
58:07 again much more complicated then this 2d
58:10 system
58:12 you get a flow diagram that looks like
58:15 this here
58:17 on the bottom so here you suddenly
58:21 have another fixed point an unstable
58:23 fixed point
58:25 in between these two extremes now this
58:28 unstable fixed point
58:30 here if you start to the left of this
58:33 unstable fixed point you were driven to
58:36 a state
58:37 without that that we had previously
58:40 where the coupling is very low
58:42 or that corresponds to the system at
58:43 very high temperature
58:45 if you start to the right of this you'll
58:47 be driven to a state
58:49 where you have order you know where your
58:51 coupling is basically infinity or your
58:53 temperature effective temperature
58:55 is zero and because you have now this
58:58 fixed point here this new fixed point
59:00 right you get
59:02 this singularity or this discontinuity
59:05 of the free energy because if you go a
59:06 little bit to the left
59:08 you go to another a different
59:10 macroscopic state
59:11 then if you go a little bit to the right
59:14 and of course you can test that with
59:15 numerical simulations
59:17 now so that's here from the book of
59:19 cardi which is a very very nice book um
59:22 scaling and renormalization and
59:23 statistical physics
59:26 and i have to say that
59:29 the class don't look very good on this
59:32 ipad
59:34 so what you see here are just
59:36 simulations of the two the eisenmann
59:39 and what they did is they performed one
59:42 block spin removalization procedure
59:45 that's that's what we did right now
59:47 that's the coarse grain so one step of
59:50 this coarse graining
59:52 and uh if you are right at the critical
59:55 point on the left hand side
59:57 you do the coarse graining step yeah
59:59 then
60:00 the system remains invariant if you
60:02 start in a fixed point you'll stay there
60:05 if you start a little bit this course
60:07 grading procedure
60:08 and here that's not there's nothing
60:10 fancier they just took the simulations
60:13 and they did one course grading step
60:15 they averaged you have maybe over a
60:17 block of spins or so
60:19 and if you are above this
60:23 critical temperature that means you
60:24 start here on the left of this fixed
60:26 point
60:27 and if you do this course grading step
60:29 then your system looks more disordered
60:31 than before
60:33 yeah so this year it looks like a higher
60:35 temperature than this year because you
60:37 have a lot of these small domains
60:39 now this is just a reflection of this
60:41 stat like an intuitive
60:43 picture of how you go in this
60:45 reunionization procedure
60:47 to the left to a state that has no
60:49 coupling at all
60:50 k equals zero and if you're coupling a
60:53 zero that's the same as when your
60:54 temperature
60:55 is very large and if you want to read
60:58 more about this
60:59 have a look at the book of john carty
61:01 and there's also of what i just showed
61:03 you
61:04 there is a there is a nice book by um
61:11 so there's a nice book there's a nice
61:13 article by karanov
61:15 um by teaching the immunization group
61:17 and he does these calculations also for
61:19 the 2d
61:20 model okay so now
61:23 we state in a real space yeah and in
61:26 real space
61:28 uh is very intuitive now
61:31 and it works for the one deising model
61:33 for the 2d eisen model gets already
61:34 complicated
61:36 and it's basically impractical to do
61:38 that
61:39 for general um
61:43 for for general physical models now it
61:46 gets very complicated to do that
61:47 procedure
61:48 in real space and the reason is that
61:50 there's no small parameter involved
61:53 that you can use for an expansion
slide 15
61:57 then there was another guy called wilson
61:59 who came up with another idea
62:01 now that was actually and that's called
62:03 the wilson
62:05 momentum shell idea
62:09 that's the world's in momentum shell
62:11 idea so what does it mean so what was
62:14 wilson's idea was that we caused grain
62:18 by integrating out fast degrees of
62:21 freedom
62:22 or degrees of freedom that have a very
62:24 short wavelength
62:26 in fourier space that's the way you do
62:29 that
62:30 is uh you look at free space also this
62:33 is our
62:34 free space let's say we have two
62:35 directions in free space
62:38 then we have here
62:44 a maximum wave vector that's the maximum
62:52 wave vector and this wave vector let's
62:55 call it
62:56 capital omega is the same
63:00 just given by that's the smallest
63:02 structure we can have in the system
63:04 that's a microscopic length scale
63:06 yeah and that's in these lattice systems
63:08 this typical
63:09 uh one over the the lattice the
63:12 microscopic level the lattice spacing
63:15 yeah so we cannot go any smaller than
63:17 that
63:19 now starting from the smallest length
63:21 scale now so a description of our system
63:24 on the smallest length scale
63:27 we now integrate out the blue stuff here
63:32 that's this one here that's the momentum
63:35 shell
63:42 and we integrate out this momentum shell
63:45 until we reach
63:48 a new wavelength a new y vector of a new
63:51 length scale
63:54 omega prime is equal to the original
63:57 omega
63:58 divided sum by some number lambda
64:01 yeah so
64:05 we integrate out one bit in momentum
64:08 space
64:10 at a time and that means that we perform
64:13 an integration
64:15 uh on a momentum shell on a tiny shell
64:18 in momentum space and also our new field
64:26 in the momentum shell
64:32 is then called typically something like
64:35 feel
64:36 find larger of q
64:40 and this is just undefined by
64:43 phi of q
64:46 with q in this interval
64:51 omega over lambda to omega
64:56 yeah so we integrate
65:00 out one step and now wilson's scheme is
65:03 actually very similar
65:08 to what we've done already yes the first
65:10 step will be
65:12 recall screen
65:18 now by rescaling oops sorry
65:29 no by rescaling
65:34 and that will give rise to some
65:38 change in the coefficients
65:46 in the action
65:49 and the second step is that we perform
65:56 this integration in the momentum shell
66:00 integrate out
66:04 short range
66:08 fluctuations
66:12 or momentum
66:19 a second step and we'll in the next
66:22 lecture
66:22 we do exactly this yeah we performed
66:24 this wilson's ruralization group
66:26 procedure that is much more practical
66:29 than the
66:29 block spin immunization group that we
66:32 had in the
66:32 beginning of this lecture and the good
66:36 thing about this wilson's
66:38 of wilson's idea is that it actually has
66:40 a small parameter
66:41 this momentum shell is very small
66:44 and uh yes so this
66:48 has a small parameter that means that we
66:50 can actually then
66:51 hope to get some approximative um
66:55 uh scheme out of this approximative
67:00 so that we can approximate our integrals
67:02 that we get from the course grade
67:04 yeah so we'll do that uh next week
67:07 for our little epidemic model and we'll
67:10 derive
67:11 the renalization group flow from our
67:13 equity epidemic model
67:15 and from this flow we'll then get the
67:17 exponents that derive that describe
67:20 the action of the the the behavior
67:23 of this epidemic model near the critical
67:26 point
67:27 yeah and um
67:30 exactly yeah so so that's what we'll do
67:33 i'll
67:33 just leave it for here today because and
67:35 then next week we do the calculation and
67:37 if you're
67:38 not interested in calculating that
67:41 because it's so
67:42 uh so uh
67:46 so uh short before christmas if you're
67:48 free to skip the next lecture
67:50 yeah and uh officially i think it's not
67:53 a lecture data
67:56 but i wanted to get that done before
67:57 christmas that we can after christmas at
68:00 january what is that fifth or so i can't
68:03 remember
68:03 um we'll start actually done with data
68:06 science and
68:07 to look at some real data yeah okay
68:10 great
68:11 so that was today only the intuitive
68:13 part about romanization so next week
68:15 we'll do
68:15 we'll see that in action and see how it
68:18 actually works in the non-equilibrium
68:19 system
68:22 bye
answering a question
68:28 excuse me yes
68:31 um could you please explain again why
68:33 were we trying to reach the fixed point
68:35 from the critical point
68:37 so so we're not trying to
68:41 it's just that you assume for this to
68:44 work that there is such a fixed point
68:47 that determines the flow of the
68:50 renormalization group yeah in this
68:53 generality you have to assume that that
68:55 there is
68:56 such a fixed point and from nonlinear
68:58 dynamics dynamical system
69:00 lecture that we have we know that once
69:02 we have such a fixed point
69:04 we basically know already how the system
69:07 behaves also in other parts
69:09 of uh of the faith yeah so that's why
69:12 the system these fixed points are so
69:14 important now we have to assume that
69:16 there exists some verb
69:17 uh but we have to basically for every
69:19 individual model that we look at we have
69:21 to show that they actually
69:23 that they that we have actually
69:25 meaningful
69:27 or moral more than one thing of course
69:31 yeah so but once you have the flow it's
69:33 a problem in normal dynamics
69:35 that's that once you have the flow you
69:36 do what you do in non-aerodynamic
69:38 dynamic dynamics so typically in these
69:40 books of linearization they use a
69:42 different language that's a little bit
69:43 disconnected from this dynamical systems
69:47 field
69:48 no but what you do is you have
69:50 non-linear
69:51 differential equations and then you just
69:53 want to see what happens to these
69:55 nonlinear differential equations
69:57 and then if you ask this question then
70:00 and non-in your system you need to ask
70:01 about the fixed points
70:03 and about the stability of yeah and
70:05 that's why this fixed point
70:07 in the randomization group flow is so
70:10 important
70:11 now if there wasn't any fixed point at
70:13 all yeah then
70:14 it would be a very good system so we
70:16 have to have such a big
70:19 point on the critical manifold for this
70:22 uh for this procedure to work
70:25 now we have to assume at this stage here
70:27 we have to assume
70:29 that it exists and if it exists then it
70:31 will determine
70:32 our flow yeah
70:36 but we don't want to we don't want to do
70:38 so once we have to fix the the system
70:40 the renewalization group will
70:41 automatically
70:42 carry us to the fixed point now we don't
70:44 we don't want to go the system to
70:47 the system to go there if the fixed
70:49 point exists
70:50 we'll have to uh we know that the flow
70:54 will determine we will be determined by
70:56 this
70:58 yeah so that's the that's the that's the
71:01 idea
71:02 but there's non-linear systems that in
71:04 principle has not much to do with
71:06 renormalization
71:07 yeah it is a general property of the
71:10 dynamic
71:14 okay thank you okay great
71:24 any other questions
71:28 um i have a question um so
71:32 does this does this um
71:35 require you to have a very
71:38 good understanding of the microscopic
71:42 um dynamics i guess
71:45 you have to have a model to start with i
71:48 mean
71:48 um but that's kind of what i mean is
71:51 if you have some model that's leaving
71:55 out certain things that are
71:57 that maybe
72:01 you know you thought wasn't so important
72:03 or something like that
72:04 but then as you core scream more and
72:08 more
72:08 like does it like um
72:11 will it give rise to like a different rg
72:14 flow than
72:15 if you had included you know other other
72:19 uh okay i think i think that that
72:21 depends on
72:22 what you leave away and what you leave
72:24 out and what you can use so in principle
72:28 for the usual model including this si
72:30 model that we have here
72:32 that we're looking at and also but also
72:34 for the icing model and
72:36 they they fall a little bit out of the
72:37 blue when somebody presents them to you
72:40 and for example in the lecture now they
72:43 completely make sense that they
72:45 destroyed these systems
72:47 but usually people have used
72:50 renormalization studies
72:52 to show that additional terms don't
72:55 don't matter for these yeah also for
72:57 this active
72:58 uh for these active systems that we
73:00 talked about
73:02 with these aligning uh with these
73:04 aligning directions
73:05 and so on now for these systems i showed
73:08 you briefly a larger way equation and
73:09 people did a
73:10 lot of work to show that this is
73:12 actually the simplest
73:14 uh description that you can have that
73:15 describes this system now because the
73:18 rigid rimmelization they found
73:20 that all other terms so this is the
73:22 minimal set of terms that i need
73:24 to describe still the same
73:26 renewalization
73:27 or still renumerization yeah
73:31 and uh whether you get new terms i
73:34 so i i would i wouldn't expect that if
73:35 you
73:37 i wouldn't expect to get new relevant
73:39 terms
73:40 out of nothing yeah in general you know
73:43 otherwise you could start with just
73:45 nothing at all and see what happens and
73:47 then you get like a theory of everything
73:49 i wouldn't expect these things to pop up
73:52 yeah but of course you get all kind of
73:53 messy things
73:55 now you can have if you start with the
73:57 icing more then in reality then you get
73:59 higher order
74:00 interactions and so on and
74:04 and you have to basically have to get
74:06 rid of them
74:08 okay i think let's uh okay great
74:18 okay so if there are no more questions
74:20 then uh
74:21 see you all next week so as as
74:25 as usual next week i'll uh
74:28 start with a repetition of this and
74:31 explain explain it again before we
74:33 actually do the
74:34 real calculation okay bye see you next
74:40 week
74:45 you