还是重整化群，冲不动了……
00:11 [Music]
00:14 hmm
00:23 i didn't expect so many people to join
00:25 shortly before christmas
00:26 i thought you were all already
00:30 traveling
00:33 so today we have a little
00:36 extra lecture yeah it's not only
00:40 christmas but it's also the lecture
00:43 where we finally got to do some
00:44 calculations some realization
00:46 calculations and uh because this is a
00:50 lecture focused on calculations
00:54 um it of a little bit of an add-on
00:58 lecture
00:59 yeah so we have the intuitive stuff we
01:02 did last time
01:04 and now we see how it works in practice
01:06 but you don't need this lecture
01:08 actually for the remainder of this
01:09 course
01:11 so let me share the screen um
01:15 so you see as you see before i switch
01:17 off the video so you see that i
01:19 brought a little uh pyramid here it's
01:21 one of the traditional
01:22 things that is produced in the mountains
01:25 around grayson
01:27 and uh that's what a lot of people now
01:30 have in their windows
01:32 or in their uh apartments
01:35 and uh there's a lot of tradition
01:38 christmas traditions in germany actually
01:39 come from this area here around dresden
01:42 and so let's
01:46 move on and let me give you a little
01:49 reminder um
slide 1
02:07 there we go let's start with a little
02:11 reminder
02:14 there we go yeah so you've seen that
02:16 already
02:17 now many times for some reason it took
02:19 us an entire lecture to just
02:21 define this model and
02:24 so this is the epidemic model we want to
02:27 treat
02:28 analytically today as this analysis this
02:31 epidemic model
02:32 consists of two kinds of people the
02:36 infected ones the susceptible ones
02:38 and then we have we put these people on
02:40 the lettuce and the world
02:42 and when uh and these infected people
02:45 can infect
02:47 uh susceptible ones or non-infected
02:49 people
02:51 if they are on the neighboring letter
02:53 side
02:54 and then infected people can also
02:56 recover and turn
02:58 into susceptible or non-infected people
03:02 so this is the little model that we
03:03 introduced and uh
slide 2
03:05 i also just a quick reminder that this
03:08 model
03:09 produces uh critical behavior that means
03:13 that we have
03:14 a value where we balance uh the
03:18 interaction and recovery
03:19 rate in a certain way uh where
03:23 uh the there's a such a balance between
03:26 between
03:27 infection and recovery and uh if we
03:29 attune this parameters to be in the
03:31 state
03:32 then we get these self-similar states
03:34 that you see in this in the middle
03:36 where both the spatial correlation
03:38 length but also the temporal correlation
03:42 is infinite yeah
slide 3
03:45 then we moved on and we uh
03:49 inferred the field theory description
03:52 of this uh s i model
03:55 yeah and the future description the
03:57 martian citra rose function integral is
03:59 here on the top
04:01 now that's the that's the margin as it
04:04 was the generating functional
04:06 and you can see here we essentially have
04:09 an order of equation
04:10 and then we get these additional terms
04:12 here
04:14 that's because we have effective
04:16 interactions between letter size
04:17 and multiplicative noise that means that
04:20 the noise
04:21 itself depends on the strength
04:24 of these feet on the concentration of
04:27 these
04:28 individuals now the function integrands
04:30 has
04:31 the function integrals has two kinds of
04:34 fields
04:35 now the five field which is the density
04:37 of
04:38 infected people but we also have the phy
04:41 to the field which is the response field
04:43 a couple of lectures ago
04:45 we discussed that this is describes the
04:48 instantaneous response of our field
04:51 to very small perturbations that's the
04:54 intuition about this response
04:57 and we get this response here because we
04:59 don't have the
05:00 noise explicitly here anymore so the
05:03 response field
05:04 uh in some way mimics the noise
05:09 yeah so we can write this functional
05:13 uh generating functional by separating
05:17 the action into two parts we are a free
05:20 part
05:21 so we call that free part because in
05:23 this free part
05:25 we have terms
05:29 that are quadratic in the fields
05:32 now so this is the gaussian part that we
05:34 could integrate if you want
05:36 now is this first part here that's
05:38 quadratic and then we have terms
05:41 that have higher order interactions
05:43 between the fields
05:45 yeah phi squared times phi tilde and so
05:48 on
05:48 yeah and these we call this we call the
05:51 interaction
05:52 part and uh we treat this interaction
05:56 part the first part is essentially
05:58 gaussian as we have second order in the
06:00 field
06:01 and we can hope that you can deal with
06:03 that
06:04 but the other parts are non-gaussian so
06:06 we have higher orders
06:08 in these fields and we don't know how to
06:10 perform integrals
06:11 around this last interacting part so we
06:14 separate these two
06:16 and if we separate this two and we do
06:17 some rescaling
06:19 you know of these parameters to make
06:21 things look simple
06:22 we get to this form where the three part
06:25 is given
06:26 as usual yeah and the interacting part
06:29 now has a more compact form that we also
06:32 already derived
slide 4
06:34 two lectures ago now we can
06:37 transform this to fourier space
06:40 and uh in this fourier space uh this
06:44 uh the spatial derivatives this one here
06:48 uh become algebraic quantities so
06:52 the wave vector squared here represents
06:55 diffusion
06:56 and here the omega i omega
07:00 is what we get from the time derivative
07:02 and then we have here
07:04 these interaction terms that you look at
07:06 as complex
07:08 as usual
slide 5
07:12 so now we want to understand the
07:16 critical behavior of this model
07:18 that means what we want to understand is
07:21 the
07:22 macroscopic behavior of
07:26 macroscopic quantities in the vicinity
07:28 of the critical point
07:30 and i told you already that in the
07:31 vicinity of the critical point just like
07:33 in the
07:34 equilibrium system that we get
07:38 divergences now so for example the
07:40 correlation length diverge
07:43 with uh power laws you know and to
07:46 understand these power laws and to get
07:48 the exponent
07:49 we have uh we could the principle
07:53 naively you know if you want to have a
07:55 system at a critical point we want to
07:57 understand the macroscopic behavior
07:59 we could naively just say okay let's
08:01 just average
08:03 over the entire system over large areas
08:05 in the system
08:06 and write down a macroscopic theory of
08:08 these averages
08:11 yeah but that's not what's working out
08:13 that's called mean field theory
08:15 that's not working out very well because
08:18 in mean field theory we
08:20 basically immediately go to the
08:22 macroscopic scale
08:23 at a critical point we have the
08:26 self-similarity
08:27 now we zoom in and the system looks the
08:30 same as before
08:32 and because we have the similarity all
08:34 length scales
08:36 are equally important they all matter
08:39 so we need an approach that allows us to
08:41 go from a microscopic scale
08:44 step-by-step scale by scale to the
08:46 macroscopic scale
08:49 and renormalization allows us to do that
08:52 we start with a microscopic theory like
08:55 our lattice model
08:57 and then renormalization
09:01 renewalization allows us to go
09:05 from the microscopic scale step by step
09:09 to the macroscopic scale
09:12 now i derive a description on the
09:14 macroscopic scale
09:16 now this renewalization has two steps
09:19 the first step was
09:21 coarse graining
09:26 that's the basic idea and with this
09:29 course gradient i showed you that
09:31 you have a lattice system for example an
09:34 ising model
09:36 that consists of lattice points
09:42 then this course graining step you can
09:44 think about
09:45 as for example defining blocks
09:49 of such spins now think about a magnet
09:52 or so
09:54 and then representing each block
09:57 by a new variable by a new
10:00 effective lattice side or new effective
10:03 spin
10:05 now the second and third steps were
10:09 rescaling
10:14 schooling and renormalization steps
10:21 now that means we need to make sure that
10:23 we when we do this
10:24 procedure of course green is essential
10:27 essentially makes
10:28 our system look fuzzy i think if i do
10:31 like this with our
10:33 with with my glasses here then
10:34 everything everything is fuzzy
10:36 and my eyes are doing cold straight yeah
10:39 so
10:39 if i do that procedure of these blocks i
10:42 now have to make sure that the length
10:44 scale
10:45 that i get in my new system corresponds
10:47 to the old length scale
10:48 so that we can compare this distance and
10:51 about this we everest rescale lengths
10:57 so that our sites have the same
11:00 letters as spacing at the old level
11:02 sides
11:04 and we also need to rescale energies to
11:07 renormalize the field
11:11 now to so that the new spins have the
11:13 same magnitude for example plus one and
11:15 minus one
11:16 as the old states now if we do that
11:19 then we go from a microscopic if this
11:23 was the
11:24 described by a microscopic some action
11:27 here
11:29 some action if we do these steps
11:33 we get a new action as prime
11:37 and if we do this course grading
11:39 correctly
11:40 then we can hope that the new action has
11:43 the same structure as the old
11:45 action that means that the old that the
11:47 new action has the same terms
11:49 in it just with rescaled
11:52 parameters now and
11:56 the the way this looks then also i also
11:58 showed you already this
11:59 picture is that if you consider the
12:02 space of all
12:03 actions p1 p2 that is described by some
12:07 parameters
12:09 then our physical epidemic model
12:14 has some space in this
12:18 has some subspace of the space of action
12:20 and somewhere in the subspace is our
12:22 critical point
12:25 now we renormalize we start with a
12:28 microscopic
12:30 critical description for microscopic
12:33 critical action
12:35 and then we normalize and ask where does
12:37 this renormalization
12:39 lead to
audio problem
14:13 uh stefan uh we are unable to hear your
14:16 audio
14:17 like it's a problem with all of us and
14:20 everyone's writing in the chat
14:22 there's some problem with the audio
15:16 okay can you hear me now
15:21 yes so now you can know you can hear me
15:24 now i'm using
15:25 the macbook microphone
15:47 okay so
15:56 so it's always killing the bluetooth
16:00 connection for some reason
16:22 [Music]
16:27 okay i have to
16:48 can you hear me
16:54 okay but i think it's the
16:57 is it correct that the quality is not
16:58 very good
17:05 no it's not working it's okay okay let
17:08 me know if
17:09 it's uh not okay because i'm using the
17:11 thing that should give you the echo
17:14 um let me know it's not okay i'm gonna
17:17 try again with
17:18 the headphones okay so i don't know
17:22 where i actually
17:23 stopped yeah uh can you let me know when
17:26 you when you stopped
17:28 being able to listen to did you hear me
17:32 uh you were talking about uh where the
17:34 si model lies and then
17:37 you started drawing the subspace and
17:38 then we stopped hearing
17:40 from the when you started drawing the
17:41 fixed point okay
17:44 okay
18:03 yeah it says strange that it's working
18:05 all the time
18:06 and then suddenly it stops working okay
18:09 so let's see
audio back normal
18:10 okay so i stopped basically
18:16 with the critical point of the si model
18:20 then we normalized and this
18:22 renormalization process
18:24 leads us to a fixed point
18:27 once we are in the fixed point the
18:29 action is always
18:30 mapped onto itself
18:34 yeah that means in this fixed point once
18:36 we are in this fixed point
18:38 or the sixth point describes the
18:40 macroscopic
18:42 properties of our system
18:47 and then i defined the set of all
18:51 actions that are also drawn into the
18:55 same fixed point
18:57 and that's called the critical manifold
19:02 pretty cool manifold
19:08 yeah and this critical manifold yeah
19:11 and then we looked at a different action
19:14 for example this one
19:17 here and if you look at this different
19:19 action that's called micro is going for
19:21 example to a different
19:22 epidynamic model and this
19:25 other action also intersects
19:29 has a critical point and intersects this
19:31 critical manifold somewhere
19:33 and when we normalize this other
19:36 microscopic theory then
19:40 the action will be drawn into the
19:42 facility of the very same fixed point
19:45 that means on the macroscopic level
19:49 both modal artists are described
19:53 by the same theory by the same action
19:56 and that's called universality
20:02 that different theories that different
20:05 systems
20:06 uh that differ on the microscopic scale
20:10 show the same marvelous microscopic
20:13 behavior
20:14 and that allows us for some of you you
20:16 know that from
20:18 statistical physics uh magnets you can
20:21 have
20:21 different ferric magnets of different
20:23 materials and they all show the same
20:25 critical behavior
20:26 and we're able to describe all of them
20:28 or many of them
20:30 with a very simple model that's called
20:32 the icing board
20:34 and that's because of this universality
20:36 on the macroscopic scale
20:38 only a few things matter and many
20:41 different microscopic
20:42 theories microscopic descriptions show
20:45 the same microscopic behavior
20:48 and the renormalization group allows us to
20:51 understand
20:51 why this is the case
20:55 so and this in reality
20:59 is also the reason why we're here
21:01 looking at such a simple model
21:04 for an epidemic epidemic is something
21:07 super complex
21:08 there's so many variables and parameters
21:11 but if you're interested in the critical
21:12 behavior
21:14 then we can show that in this action
21:17 that we have
21:18 if we added more and more processes to
21:20 it more and more turns more
21:22 interactions but at least interactions
21:24 in many cases
21:25 are much relevant on the macroscopic
21:28 scale
21:30 and then the next step you can take this
21:32 fixed point
21:33 and calculate exponents and we do this
21:36 by looking
21:36 by looking into the directions that
21:39 drive us
21:40 away from the critical manifold these
21:43 are the relevant
21:46 directions
21:51 yeah and if we ask how fast the systems
21:54 these are the parameters that
21:56 experimentalists needs to tune
21:58 in order to bring the system to the
22:00 critical point
22:02 and if you ask how quickly
22:05 is the renovation flow driven out
22:09 of the critical point or talking about
22:12 to the critical manifolds
22:15 then this describes uh then we can
22:18 derive
22:18 exponents from this exponents
22:22 tell us how fast the system goes out of
22:25 the critical manifold
22:27 once we renovate so that's the
22:30 general idea now let's see how this
22:33 looks in practice
22:34 we started already as another another
22:38 reminder is the wilson's renovation
22:41 momentum shell renovation
slide 6
22:44 wilson's idea was that what i showed you
22:46 on the last
22:47 slide here is these blocks the problem
22:50 with these blocks is that you
22:54 there's no small parameter you cannot
22:57 make an epsilon block or something like
22:59 this
23:00 yeah so that's that's that's a that's a
23:03 problematic thing
23:04 and that's why these blocks also called
23:07 real space criminalization
23:09 is very often very often doesn't work or
23:11 is very difficult
23:13 so wilson idea wilson's idea was to do
23:16 the regularization
23:17 in momentum space and that's also a
23:20 reminder
23:22 because we had that already last time
23:24 what we do
23:25 in wilson's momentum shell
23:27 renormalization
23:29 is that we take a look at the space of
23:32 all
23:32 wave vectors and integrate out
23:37 uh the smallest wave vectors
23:40 and so again as before there's like two
23:43 steps to rescaling
23:44 the ends a realization and the cause
23:47 training and here in this case we do the
23:49 course training
23:50 by integrating out a tiny
23:53 shell in momentum space we integrate out
23:56 the fastest wave vectors which
23:58 corresponds
23:59 to the shortest length scales
24:04 and then we can formally
24:07 describe our fields as fine for example
24:12 is the component by a short wavelength
24:16 plus
24:25 the slower wave vectors and the fast
24:27 wavelengths rather we integrate out
24:29 these
24:30 fast wave vectors at each steps
24:33 at each step
24:37 yeah that means that we define
24:41 our action on the
24:45 long so small wave vectors with the long
24:48 wavelengths
24:49 as the integral over the entire reaction
24:53 but we only perform this integral
24:57 over the very this momentum shell
25:01 on this very highest wave vectors
25:04 in the system yeah and if we integrate
25:07 this out
25:08 now that our momentum shell our momentum
25:11 space gets smaller and smaller
25:13 until we arrive at momentum
25:16 zero that corresponds with very small
25:20 values of these skews that corresponds
25:23 to very large length scales and so
25:24 therefore
25:25 macroscopic behavior
slide 7
25:29 okay so let's begin
25:32 so we begin we begin
25:37 by doing the rescaping stuff and this
25:40 lecture is different from mass vectors
25:41 but this will
25:42 probably mostly be a chalk chalkboard
25:46 lecture
25:46 and we'll have to see how this works on
25:48 an ipad
25:50 and so so let's let's see i hope it's
25:53 not too confusing
25:57 okay so first we do rescaling
26:06 and is rescaling
26:09 to say that we have to
26:13 rescale our length skills
26:17 by some
26:21 like london
26:25 with the value of longer than smaller
26:26 than one
26:31 if we rescan our length skills we also
26:34 have to rescale all
26:35 other things the answer for length here
26:39 goes like this so our x and our action
26:44 the time
26:48 also needs to be scaled that's not
26:51 independent
26:52 of space because it connects to space
26:55 by this dynamic exponent z
26:59 yeah that was the ratio between the
27:01 perpendicular and the parallel
27:04 correlation exponents um
27:07 so that's not independent of space we
27:09 have to reschedule as well
27:11 and we don't know what that is by the
27:13 way but that's our goal
27:15 and then we have to escape our fields
27:19 what our fields we scale with some
27:21 exponent
27:23 chi when we
27:26 rescale our x and our
27:30 time this way
27:34 and our other field
27:38 now the volatility we scales in the same
27:42 way
27:43 5 tilde lambda
27:47 x lambda z
27:51 t yeah
27:54 so we don't know what chi is now we
27:57 don't know what
27:59 that is but if we rescale the length we
28:02 have to rescale the other things as well
28:06 yeah and kind at five
28:09 sorry fine
28:12 and fragile have the same scale
28:16 because uh if we replace
28:19 one by the other and we switch time to
28:22 the action
28:23 and then we get the same action back so
28:25 these 5 and 5 total fields
28:27 are the same thing if we transform the
28:30 action
28:31 accordingly okay so now we just plug
28:35 this
28:35 in into the action with this
28:39 we get that as not
28:43 let's find phi together
28:46 we have these integrals here d dx
28:52 dt
28:55 by children of x t
29:01 and then comes this part
29:04 of tom times
29:08 lambda two
29:12 coin plus b where does that come from
29:15 possibly data
29:16 delta t so we have here
29:21 one length scale that gives us the d
29:25 lambda to the power of d
29:28 we have a times the integral over time
29:31 that gives us uh that cancels out with
29:34 this one that is one over time
29:36 this time so we don't have anything and
29:39 the two kai
29:40 can't because we have the kai from the
29:42 the five the field from the left hand
29:44 side
29:45 and later they flew from the right hand
29:46 side
29:48 so now we have this part
29:52 minus d
29:55 lambda 2 chi
29:59 plus d plus z minus 2
30:04 times
30:07 uh sort of minus
30:11 here it's again this this comes from the
30:13 fields the two fields
30:16 this comes from the uh
30:20 sorry
30:23 so this comes from the from the integral
30:26 of a space
30:28 this comes from the integral over time
30:31 and the minus 2 comes because this was
30:36 originally a second derivative in space
30:39 we had a diffusion trip here so that's
30:42 how we get this
30:46 so minus pepper lambda
30:50 [Music]
30:55 phi of x t
30:58 so now i just assume we did the
31:00 resetting step
31:02 just zoomed out and this
31:05 already gives us some change of
31:08 parameters
31:11 and we can now call these parameters
31:15 give them new names for example
31:19 this one here is now tau prime
31:24 this one is d prime
31:28 and this one is kappa prime
31:34 now so this was the the part of the
31:37 action at this
31:38 second order now we take the part of the
31:40 action that has higher organisms
31:47 and this part is also an integral dd
31:50 x integral dt
31:55 gamma that is the strength of the voice
32:13 in third order
32:18 to coin plus a from the integral
32:22 over space and plus z from the integral
32:25 over time
32:26 now
32:32 of x t times
32:37 y of x t minus phi tilde
32:43 of x t
32:46 also you see that these fields
32:50 always appear that cubic order
33:05 yeah so now we have already
33:08 an equation that updates
33:11 our parameters by this one step
slide 8
33:15 right so now we say that for
33:21 [Music]
33:22 infinity
33:24 small uh coarse graining
33:31 that means our momentum shell is very
33:34 small
33:35 so we said that that this capital wonder
33:38 is something like one
33:40 plus l
33:46 we obtain
33:50 the first order
33:54 the following updating scheme tau
33:57 prime is equal to
34:00 one plus that's of course the taylor
34:03 expansion
34:04 l times two chi plus d
34:10 times tau
34:14 we have copper prime
34:17 sorry x one is d
34:20 this d prime
34:21 [Music]
34:23 one plus l two comma plus
34:27 d plus z minus two
34:34 [Music]
34:37 prime is one plus
34:41 l two pi plus
34:45 d plus z times copper
34:49 and gamma prime is one plus
34:53 l three chi plus
34:56 d plus z
35:00 times so that's the updating
35:04 of our parameters based on the restated
35:07 steps
35:08 now and this updating of these
35:11 parameters
35:12 is a result of that is dimensional
35:14 [Music]
35:15 same principle you can look at this you
35:18 can get these
35:19 updating just by looking at the
35:20 dimensions of things
35:25 and so this part here
35:28 was mathematically so let's say zero
35:32 effort
35:33 but in the second part the course
35:36 burning part
35:37 you have to invest a little bit more
35:39 thought
35:42 so how do we do that
slide 9
35:46 no so how do we do the cosplay instead
35:48 cosplayer stuff
35:49 is difficult because we have to
35:51 integrate
35:54 over the momentum shell in this action
35:57 and this action
35:59 has parts that are cubic in the fields
36:03 that we don't know how to integrate
36:06 we know how to perform gaussian
36:07 integrals we have second order terms in
36:10 the fields
36:11 but we don't know how to integrate third
36:13 order terms in the field of higher order
36:15 terms
36:17 yeah so think about uh also like
36:20 statistical physics five to the four
36:22 terms
36:23 in the fight to the lambda we have a
36:26 five to the fourth term
36:28 and we don't know how to integrate these
36:30 things
36:31 so what we do is
36:34 what we say so and only so these
36:38 these calculations are really empty what
36:41 i'm
36:42 doing here i just give you for this
36:45 course grading stuff i just give you
36:46 sort of an overview of the steps
36:48 but i don't perform the actual integrals
36:51 now if you're interested
36:53 in the details
36:57 there is a review by hindelison
37:04 and this review is about
37:10 non-equilibrium
37:14 what is it molecule or phase conditions
37:16 or some non-including
37:18 something and but you immediately find
37:21 it because there's not
37:22 so many people who are archimedes and
37:25 write revenues with
37:26 uh 1 600 citations
37:30 uh on one equilibrium phase positions
37:33 and there you can find in the appendix
37:35 all of these calculations and how to
37:38 perform these calculations
37:40 so i'll just give you uh
37:44 a glimpse of how this works so the first
37:48 step is to say
37:49 okay we expand our action
37:52 to first order and that's called the
37:56 cumulative expansion we do this
37:59 accumulated
38:01 equivalent expansion then we see that
38:04 our
38:04 next action our updated action
38:08 is equal to the action
38:15 in the double wave vector so on the
38:17 larger length sets it inside
38:19 the momentum shell so in the core
38:23 of mental space plus
38:26 the first moment of this interacting
38:30 action
38:33 so plus the average expectation value
38:38 of this interaction part of the action
38:42 evaluated for the in the momentum shell
38:48 and evaluated in the context of the free
38:53 action that only has the gaussian term
38:56 so that's
38:57 what we see what we see here then the
38:58 higher order terms
39:00 so this contribution that we get is
39:02 integral
39:05 where the action here
39:10 so the definition of this average
39:13 is like this that we integrate only over
39:16 the
39:17 momentum shell so the very the highest
39:20 the highest momentum we have
39:21 in this in the system uh and we
39:24 wait for the for the average weight not
39:27 by the full action
39:29 but by the gaussian action
39:32 now that's the approximation then there
39:34 are higher order terms
39:35 that come on top of that and so what you
39:39 see here is a little bit what happened
39:40 here
39:41 is that we expanded the exponential in
39:44 this
39:45 action here by assuming that these
39:48 [Music]
39:50 interactions are actually weak now that
39:52 we expand the exponential
39:54 and we have a linear term here in front
39:57 of that
39:58 and if these interactions are weak
40:02 then we can make this expansion here and
40:05 we
40:05 for now which for today we truncated
40:07 after the first order and the whole
40:10 problem reduces
40:11 to calculating this thing here
40:15 or to calculating this thing
40:19 now we still don't know how to calculate
40:21 it
40:22 but it involves something that is second
40:25 order in the fields
40:27 yeah so that might actually be something
40:29 that we can do
40:30 and there's a theorem that helps us
40:33 helps
40:33 uh helps us to do these things let's go
40:36 to rick's
40:38 so this theorem tells us that if we have
40:41 such an
40:41 average so for example this is some
40:44 product of some fields here
40:48 and if we have an average of a product
40:50 of fields
40:53 and if we evaluate this average or if we
40:56 calculate this average
40:57 in the framework of the gaussian or the
40:59 second order
41:01 action then this complicated thing
41:05 can be expressed by a sum
41:08 over all pairwise contractions
41:11 of these fields yeah that means that we
41:16 look at all pairs of the things that we
41:18 have on the left hand side
41:21 put them together into groups of two
41:25 average around them
41:28 yeah averaged around them yeah
41:32 and then and then multiply and sum
41:35 over all possible ways of how you can
41:39 partition this product here in the
41:41 groups of twos
41:43 now so for example the bottom if you
41:46 have four
41:48 fields or fields evaluated at four
41:51 point in times so here's signify by phi
41:53 one five two five eight
41:55 three five four then we just have to
41:58 look at
41:59 all possible combinations of how to
42:03 make groups of twos out of these four
42:07 fields that we have yeah so
42:10 that reduces the problem
42:14 of calculating something that is highly
42:16 familiar
42:18 to something much more simple
42:21 we only have to write down all possible
42:24 combinations of these pairs called
42:26 contractions
42:27 of pairwise pairwise these pairwise
42:31 contractors here
42:32 those are these groups of twos and
42:36 just have to sum up all possible ways of
42:38 how we can do that
42:40 yeah and each of these states here is
42:43 now an integral sorry
42:44 there's one thing i forgot that should
42:47 have a zero here
42:48 as always in the context of this
42:51 simplified
42:52 action
42:55 now so that reduces the complexity of
42:58 the problem
43:00 to something that is only second order
43:02 in the fields
43:04 from something in general and
43:07 what we don't get then is what we have
43:10 to pay for
43:11 is a bookkeeping problem because if you
43:14 can imagine right
43:16 if this thing on the left-hand side is
43:17 sufficiently complex
43:20 then what you have on the right-hand
43:22 side involves a lot
43:23 of different terms so yeah that's a
43:26 bookkeeping exercise
43:28 i mean because it's a bookkeeping giving
43:31 exercise
43:32 that's difficult to overview because you
43:34 get many
43:35 if you have like one more four terms but
43:38 eight terms
43:39 the many possible ways of how you can
43:41 make these pairs of tools
43:43 yeah and this is
43:47 why we people invented
43:50 a graphical language of how to represent
43:54 these terms and this graphical language
43:57 in statistical physics
43:59 or in quantum frequency is called fame
44:01 and diagrams
44:03 in our case this graphical language is
44:06 actually pretty simple so we also have
44:09 famous
44:09 diagrams that we place that describe
44:14 such terms here but in our case these
44:18 diagrams have a
44:19 real meaning that is connected
44:22 or an intuitive meaning that is
44:24 connected to the time evolution of the
44:25 system
44:27 so first what they found with the
44:29 derivative say
44:31 is that if you have something
44:35 if you have in terms of this structure
44:37 here
44:38 then each of these here
44:42 the answer that we've also called the
44:44 propagator that we already had
44:46 in the action so these propagators here
44:48 these three propagators
44:53 they all become like a line
44:58 yeah and this also becomes a line
45:01 and once you have things
45:04 that are integrated over
45:08 for example you have here integral over
45:11 the coordinate set that the same
45:13 coordinate appears twice
45:15 in your turn then you have to connect
45:18 these things
45:20 these legs it is fine in that one
slide 10
45:23 so we won't do that here today actually
45:26 that's
45:26 that's that's the subject of the quantum
45:28 fields just say
45:30 just say that there exists this
45:32 graphical language
45:34 now in our case because we only go to
45:37 first order
45:38 uh we don't have to deal with that but i
45:40 just want to say that these feminine
45:42 diagrams that popped up
45:44 in this uh that usually pop up in
45:46 quantum field theory
45:48 have a very nice intuition in these uh
45:51 directed percolation problems
45:53 so here it's actually i
45:57 copy they copied that from the from the
45:59 revenue of henryson
46:01 uh you
46:04 these diagrams for example look like
46:07 this one this loop here
46:09 corresponds actually to trajectories
46:11 that you have
46:13 in space not something like this now
46:16 basically what you do is you have your
46:19 building blocks
46:20 of your theory you are for example the
46:22 free propagator
46:24 now that would just be the gaussian term
46:27 and but you also have this other
46:29 building block that corresponds to
46:30 higher auditors
46:32 and then you just put them together and
46:36 these higher order terms of example this
46:38 one here have a real meaning in this
46:40 theory
46:41 in this framework because for example
46:43 this one would correspond
46:45 to a branching event or this one would
46:48 correspond
46:49 to a coalescent event
46:51 [Music]
46:52 where one of these if you look at this
46:57 space-time cross that we previously had
46:59 the upside down
47:01 then um that this would
47:05 correspond to events versus a separate
47:08 goblets form and then emerge again
47:10 so that's that's how these diagrams are
47:13 interpreted
47:14 in the frame of directed combination
47:18 now but as i said it's just a side
47:19 remark and we don't actually need that
47:21 today
47:23 because what we get is not that
47:25 complicated
slide 11
47:26 what we do today is
47:31 that we
47:35 now proceed with the
47:39 integration of this uh
47:42 of the different parts of our action
47:46 so using what we said two slides away
47:50 is 274 this formula here
47:53 so that we expand our reaction in this
47:55 frame
47:57 and wix theory
48:01 okay so let's begin
48:04 so this is now the second step is the
48:08 course training
48:12 also so let's let's say let's say course
48:14 training
48:18 integrating
48:21 out
48:22 [Music]
48:24 the short leg scales
48:29 on the momentum shell
48:36 and we start
48:40 by looking at the free propagator
48:45 i'll show you now how this looked like
48:54 what the slides go so here we have the
48:57 action and momentum space
48:58 and the free propagator it's just
49:01 what is in between uh
49:05 this is what is between the inverse of
49:08 what is between
49:09 and between the the second order part
49:13 of the action so that's the same as a
49:16 quantum p theory of if you have already
49:18 quantum
49:18 theory or statistical fluid theory uh so
49:21 that's
49:22 just the very same thing so this one is
49:24 called
49:25 the inverse of the free propagator it's
49:28 called free propagator because it gives
49:30 you
49:30 the propagator that means how you uh go
49:33 from one
49:34 point in space and time to another point
49:37 in space and time
49:39 using only the frequent theory without
49:42 interactions
49:45 so this is the free propagator and this
49:48 term here is what we have to
49:54 integrate first
50:01 okay so the free propagation was that
50:04 this
50:05 do not okay
50:08 omega that was just defined
50:12 by the okay we actually use k
50:16 [Music]
50:20 we'll just check
50:24 okay okay
50:29 dk squared is just the definition
50:32 um
50:35 minus kappa minus i tau
50:38 omega
50:41 now to
50:46 first order
50:52 the propagator
50:56 is we normalized
51:02 by formulating so what we actually have
51:06 is the inverse of this
51:07 one over this
51:14 prime prime is the pre-propagated after
51:17 the first step
51:20 also
51:34 mathematical minus
51:38 gamma squared over 2 and the integration
51:42 over this momentum shell
51:46 dk prime e omega
51:59 plus omega times
52:11 minus omega prime so now you see we have
52:14 these two
52:15 propagators these propagators
52:19 you see here and that's what we had in
52:23 the quick symbol
52:26 is basically phi
52:29 of k
52:32 phi of k prime now these kind of things
52:36 right so now we let's just speak theory
52:38 and in the same
52:39 language this is just
52:48 uh this guy so we have these two arms
52:52 here and we integrate
52:56 over the case so
53:03 so that's why they have a loop
53:11 don't get distracted by these diagrams
53:14 it's not so
53:15 not so important it's just for those of
53:16 you who have already had
53:19 quantum theory just to see the
53:26 connection okay
53:28 [Music]
53:30 and now we can write that
53:36 explicitly so this first term is
53:40 k prime prime minus value d prime prime
53:44 k squared plus i prime prime
53:48 omega now let's adjust this on the left
53:52 hand side that's how we define that it
53:53 should have the same form as before
53:57 that's equal to k prime minus
54:00 d prime k squared plus i
54:05 tau squared minus
54:09 what now comes this integral here
54:13 and i'm just telling you the solution of
54:15 course the intervals
54:17 you know to say diagrams of everything
54:19 are just a way of writing things down
54:21 but in the end you have to solve
54:22 integrals yeah and you can if you're
54:24 interested in how this works
54:26 now you can look into the appendix of
54:29 this revenue of hinduism
54:32 i show you here just the results because
54:34 what we actually want to focus on is
54:38 integral integration techniques done so
54:41 i'll just give you the result
54:42 l k d i'll tell you about this what is
54:46 omega to the power of d
54:52 two tau
54:56 one over omega
55:00 squared d minus comma minus
55:03 omega square d over
55:08 4 omega squared
55:11 d minus kappa
55:15 squared k squared
55:21 plus i tau
55:24 over 2
55:29 omega squared d minus
55:32 kappa squared omega plus
55:36 higher volatility so
55:40 this kd here
55:43 is just a surface area
55:49 of surface
55:52 area of
55:56 the domain
55:59 genome sphere
56:02 yeah and that just comes from the
56:04 integration by just integrating
56:06 over a mental shell that's what you
56:08 expect to get some kind of
56:10 seriousness of the sphere but now the
56:13 important thing
56:14 is that we don't have three different
56:15 troops i'll just
56:17 mount them in colors
56:21 we have this term we have
56:25 the d term
56:28 and we have the i
56:31 tau prime prime
56:35 double beta and now we have the same
56:38 term
56:38 on the right hand side as you have the
56:42 capacitor that pops up
56:46 here and here again we have
56:49 the d that pops up here
56:54 and here so that's the prefecture of
56:58 this k squared and we have the
57:03 tone that we have
57:06 here
57:09 and here
57:14 and this should have on the ground okay
57:18 okay so now we have kappa squared on the
57:21 left hand side
57:22 and we have terms on the right hand side
57:24 that look exactly
57:26 in structure like the ones on the left
57:28 hand side
57:29 and now we can compare them one by one
57:32 these terms and these two
57:35 this one and these two
57:39 and get our
57:42 prime prime d prime prime and target
57:46 by comparison to the right hand side
57:50 so i'll just tell you
57:54 the result that's the
slide 12
57:59 renormalization
58:03 of the model parameters
58:09 okay so we have tau prime prime
58:12 is equal to tau prime minus
58:16 well that is right both
58:20 it was just comparing the left and the
58:22 right hand side of this equation
58:24 and putting terms together that have the
58:26 same
58:27 uh the same structure
58:31 okay d over
58:35 8 omega squared d minus
58:38 kappa squared
58:42 d prime prime is equal to d
58:45 prime minus gamma squared
58:49 l k d omega squared
58:52 e over
58:56 16 omega squared d minus
59:00 kappa squared and
59:03 final one is k squared
59:06 twice is equal to
59:11 minus down prime l
59:15 kd over
59:19 four tau omega squared
59:22 d minus
59:25 and for convenience we define this one
59:30 here
59:33 now as a
59:37 and we define this one here
59:45 sp
59:57 so now we have three in the updating
60:00 scheme
60:01 total updating scheme of three of the
60:03 parameters
60:05 that in principle allows us to link tau
60:07 prime prime
60:09 so after the two or the three steps of
60:11 the minimization group procedure
60:14 we update our parameters
60:18 uh in this form here and we still have
60:20 to plug in
60:21 the tau prime from previously from the
60:23 rescaling step
60:25 but there's one parameter missing and
60:27 that's the ugly one
60:28 now that's the one that describes our
60:30 higher order terms
60:33 you can imagine these
60:36 integrals of the higher order terms are
60:40 bonds more that's beautiful
60:44 but i'll just tell you the results
60:47 mainly that the cubic
60:54 sorry
60:58 the cubic so-called vertices
61:04 that's just cubic terms in the fields
61:08 we normalize
61:12 as
61:15 the proton is gamma prime
61:19 that we have to we have these diagrams
61:23 here
61:28 this one here and
61:33 this one
61:39 this one here now so you can see that
61:41 these are interaction terms that's third
61:43 quarter
61:44 that's why they have three legs but
61:47 they're also connected
61:48 here yeah and uh but both of them
61:52 one of them is just a time reversion
61:54 reversion of the other one
61:56 so both of them have the same value
61:59 so let me just now tell you the result
62:03 of this step here gamma prime prime
62:09 is gamma prime minus l
62:12 gamma to the power of 3 k
62:16 d over
62:19 two tau omega squared
62:23 d minus kappa squared
62:28 now so now we have the updating of all
62:30 of these parameters
62:35 and what we now do
62:38 is that we go to the limit
62:42 of so so this is not very convenient
62:45 right because we have to
62:46 still to plug in the tau prime and d
62:48 prime and
62:51 then we don't have to we still don't
62:52 have anything
62:54 uh that we can deal with so we don't
62:57 know how to deal with these updating
62:59 schemes
63:00 it's much more convenient to have
63:01 something that gives a differential
63:03 equation
63:04 you know but it's easy for us to derive
63:08 a differential equation
slide 13
63:10 mainly we just set the limit
63:14 in the limit
63:17 l to zero so that we
63:21 really just integrate out a tiny bit
63:24 each step so then the momentum shell is
63:28 small
63:30 we can write
63:33 we can
63:37 write these
63:42 relations in differential form
63:52 yeah and i'll just give you the result
63:56 delta l tau
64:00 is equal to tau times two k
64:04 plus v minus two times
64:08 a and then the a was the thing but for
64:10 the integral
64:16 [Music]
64:23 [Music]
64:25 to coil plus b plus z
64:28 minus a
64:33 and then the pepper
64:38 is equal to whether the
64:41 scale derivative of color is
64:46 2 plus d
64:49 plus z minus b
64:53 and gamma interactions
64:57 are given by gamma times
65:00 3
65:04 minus 8 a
65:08 this is not called
65:11 the realization flow
65:16 this is what i showed you at the
65:17 beginning of the lecture where i said
65:19 okay so we have this space of all
65:21 possible actions
65:23 our minimalization brings us
65:26 lets us travel place through the space
65:29 of all possible actions
65:34 okay now that's the realization group
65:37 workflow
65:38 and now we're in the framework of
65:41 bonding and dynamics
65:42 and number we have some differential
65:45 equations
65:46 that are coupled and these differential
65:50 equations
65:52 we now need to treat with the tools
65:56 of non-linear dynamics
66:00 okay so first we simplify that a little
66:02 bit
66:06 and what we do is when we do the course
66:08 grading
66:10 in space and time anything that we
66:12 should get
66:15 should be independent of the scale of
66:17 courseware because our system is
66:18 self-similar
66:20 yeah and so that means we have to
66:24 we have a choice to set the length scale
66:27 and the time scale to whatever is good
66:31 for us
66:32 and i'll tell you what is good for us
66:35 we set a time scale
66:36 [Music]
66:39 set the time scale
66:44 such that dell
66:47 tau is zero
66:53 and length scale
66:59 such that dell
67:04 is it d is equal to zero
67:10 and then we get
67:13 two equations from that by just by
67:16 setting the left-hand side to zero
67:18 it's four minus epsilon which are
67:20 everybody fine
67:22 plus two chi b minus two a
67:26 is zero and two minus
67:30 epsilon plus two chi
67:33 plus z minus a
67:39 is equal to zero
67:42 yeah and here i set
67:45 epsilon equal to 4 minus d
67:56 and with this
67:59 i'm left with two flow equations
68:03 one for copper and one for
68:08 gout
68:11 copper is 2 plus
68:14 a minus b
68:18 gamma x1 over 2
68:22 minus six eight
68:27 so that that looks already a little bit
68:29 nicer
slide 14
68:35 okay so the next step
68:39 we're always almost done with the
68:42 hot stuff in the next step
68:52 next step we're interested in the fixed
68:54 point
68:55 we've got the fixed point determines our
68:57 microscopic
68:58 behavior okay so behavior
69:06 near
69:11 the fixed point
69:19 where by definition of the fixed point
69:22 cover
69:23 the derivative of capital gamma
69:27 about equal to zero
69:31 what we then get is the value of
69:35 a star that a
69:39 our complex term that we had before at
69:41 the fixed point
69:42 takes the value of epsilon over 12
69:46 and b takes the value
69:50 2 plus epsilon over 12.
69:56 and now we substitute that
69:59 into let me see
70:04 this equation here
70:18 we substitute into this equation and we
70:21 get
70:21 our first two critical exponent
70:28 exponents
70:31 chi is minus two
70:34 plus seven epsilon divided by twelve
70:40 let's say it is equal to 2 minus
70:44 epsilon divided by 12.
70:49 thus we have our first two critical
70:50 exponents
70:54 now
70:57 as a
71:01 we substitute just this into the proper
71:04 definition of the fixed points
71:06 our a's and b's are something
71:07 complicated
71:11 and uh we'll just write it down
71:14 substitute
71:18 definition of a
71:21 and b and what then
71:33 squared epsilon divided by
71:36 24 plus epsilon
71:41 so everything i'm doing right now now is
71:44 not
71:44 complicated mathematics that's just
71:46 algebra
71:49 gamma star is 2
71:52 d 24 plus epsilon
71:56 over 24 plus
71:59 5 epsilon
72:03 epsilon tau over
72:08 kd
72:11 okay so this is our fixed point and the
72:14 next step
72:16 we linearize around our phase point
72:23 linear rise rg flow
72:29 around fixed point remember
72:32 a little non-linear dynamics what we do
72:35 is we
72:36 look we put ourselves into the fixed
72:39 point
72:41 so now we've got the fixed point now we
72:42 want to say is it stable or is it
72:44 unstable
72:45 now will we be pushed out of the fixed
72:47 point or will be
72:48 sucked in is it attractive or not
72:52 now and the way we do that is we look go
72:54 into the fixed point
72:56 and what we said this dynamical systems
72:59 lecture
73:00 is that we then look at the derivative
73:03 1d system the derivative of this fixed
73:05 point and here
73:06 what we do is linear wise around the
73:08 fixed point and then the
73:10 derivative in higher dimensions is
73:12 called jacobian
73:13 now that's what we do just it's just an
73:16 expansion
73:17 around the value of this point
73:20 and what we get is l in
73:23 vector form kappa gamma
73:29 is equal to
73:32 that's the jacobian
73:35 of our flow equation also 2
73:38 minus epsilon over 4 0
73:42 0 minus epsilon
73:47 and then here the distance to the fixed
73:49 point copper star
73:51 minus copper and gamma star
73:55 minus gum and of course we have higher
73:59 orders
74:02 so the eigen values of this
74:07 jacobian they tell us whether this fifth
74:11 bond is stable or not
74:13 so here we're just looking at a
74:14 non-linear dynamical
74:17 system but we use the same tools yeah
74:19 and and if you have
74:20 not just one dimension but two
74:22 dimensions like here
74:24 we're now looking at jacobian and then
74:27 the eigenvalues of this jacobian we're
74:30 now looking at the slope
74:31 in the fixed bond as an 1d and
74:33 simplified
74:34 now look at the iron bonds
74:39 this is the
74:47 okay the eigenvalues
74:54 determine
74:58 stability
75:04 so we have that 2 minus epsilon over 4
75:09 is larger than 0
75:14 that means that the fixed point
75:20 is unstable
75:24 in the direction of the parameter cover
75:30 minus epsilon sorry that's not a real
75:32 epsilon here
75:39 epsilon minus epsilon
75:43 is smaller than zero that means the
75:46 fixed point
75:50 is
75:58 and what this means if you think about
76:00 our
76:03 language that we introduce at the
76:05 beginning of the lecture
76:06 is that the parameter kappa draws us
76:10 away from the critical manifold
76:13 and the parameter gamma pulls us
76:16 here basically pulls us into the
76:18 critical
76:19 into the relationship
76:24 that's another requisition to draw that
slide 15
76:28 so we can have a little diagram
76:31 it looks like this
76:40 and so here we have our fifth point
76:44 and that will flow
76:50 lines
76:54 our flow will go into the fifth point
76:58 along the gamma direction
77:07 and out of the fixed point along the
77:09 copper direction
77:13 that means lots of points
77:20 points on blue line
77:26 flow into the fixed point
77:30 that means we need
77:38 to tune cathode
77:42 to reach the fixed point
77:51 so now we get the final response
77:56 now with
78:00 the definition of physics
78:06 this time
78:09 was this at the very beginning we
78:11 introduced the kai
78:13 as how the fields we scale
78:17 when we change the length scale and by
78:20 this definition
78:21 of coin this is equal to our older
78:24 definition of these
78:25 problems better over
78:29 new perpendicular
78:36 [Music]
78:37 that was defined as
78:40 the dynamical critical exponent as new
78:44 parallel over new of a new
78:47 perpendicular and kappa
78:53 is our distance to the critical point
78:58 lambda minus lambda c and that's how we
79:01 call it
79:06 okay and then we just plug these things
79:08 in and we get these three exponents
79:10 yeah beta is equal to one minus epsilon
79:14 over six
79:16 new perpendicular is equal to one half
79:19 plus
79:20 epsilon over 60 and
79:24 new parallel is equal to 1 plus
79:29 epsilon over 12. and these are our
79:35 exponents now that we got from the
79:39 linearization
79:40 procedure
79:43 so how general
79:46 are these results these exponents took
79:50 a simple epidemic model and derived
79:52 exponents
slide 16
79:54 at the beginning of this lecture i told
79:56 you something about universality
79:58 now that different models are different
80:02 microscopic theories
80:04 are described by the same macroscopic
80:06 behavior
80:09 so and this is something that's not
80:11 completely understood
80:14 but the models that are disqualified by
80:16 the same critical exponents
80:18 are says that they belong to the
80:21 directed
80:22 preparation class
80:26 and the model
80:31 belongs to
80:34 the directed
80:39 percolation
80:44 universality
80:48 class if that's the so-called directed
80:51 percolation conjecture
80:56 this base phase transition
81:01 it displays
81:05 the face transition
81:12 between
81:14 active and
81:21 absorbing
81:24 phase so this existence of one absorbing
81:28 point
81:29 is very important the second thing
81:32 is after the adobe point was when the
81:35 disease got extinct the second is
81:40 order parameter the order parameter
81:48 is positive
81:52 now the system is a one-dimensional
81:54 system
81:57 so the spatial dimensions uh changes as
82:00 you see from the exponents
82:02 [Music]
82:08 the order parameter is one-dimensional
82:10 that's the other which is one
82:11 one-dimensional parameter this the whole
82:14 parameter is scalar so it's not spatial
82:18 okay the third one is
82:23 there's no other bells and whistles so
82:26 you have no
82:27 special attributes no
82:32 special attributes
82:39 like spatial
82:42 heterogeneity
82:48 yeah so if for example the infection
82:50 rate depends
82:51 on where you which letter side you are
82:53 on
82:54 then these exponents could be different
82:57 difficult they are different
82:59 so what they said is if these three
83:01 conditions
83:02 are fulfilled you can't expect your
83:05 system
83:06 to be in the directed population in
83:09 reality class
83:11 and to have the same critical exponents
83:16 now just uh before we all go into
83:18 christmas
83:20 there's now a little uh final
83:23 reveal for you yeah so
83:26 in the beginning of the lecture i
83:29 we talked about what is a
83:31 non-equilibrium system
83:34 and the way we defined it different ways
83:38 to define it
83:39 the way to define it the way we defined
83:41 it is that we said
83:43 okay the system has a contact with
83:46 different paths
83:49 and these bars are incompatible
83:53 so what are the paths in direct
83:56 percolation or in this epidemic model
84:02 normally only if anybody was in the room
84:04 now we would
84:05 try to solve that together uh but as
84:08 you're all sitting
84:10 in front of your computer and maybe
84:12 watching netflix
84:13 in parallel yeah so i'll give you the
84:16 answer
84:17 so what is actually here the bath
84:20 directed percolation
84:21 so so it's actually uh
84:24 it's actually quite difficult to see
84:27 that what are the heat
84:28 what are the paths what drives direct
84:32 percolation out of equilibrium
84:34 it's the absorbing point now that you
84:37 have a point
84:38 where you can go in but it can never go
84:41 out
84:42 and when you're in that point then
84:45 you're clearly not an equilibrium
84:47 because there's no terminal there are no
84:49 terminal fluctuations
84:52 now in reality so and this is an
84:54 approximation
84:56 that you have an absorbent state in
84:58 reality
84:59 you can get out of the absorbing point
85:02 you just have to wait a few hundred
85:03 million years
85:05 for the virus one that had gone extinct
85:09 to come back by evolution that takes a
85:11 long time but this process exists
85:13 but we say in this theory that
85:17 this probability of this rate which you
85:19 get out of this
85:20 point out of the absorbing state is
85:23 exactly equal to zero
85:26 now that's the tiny thing that we do and
85:30 what this means is that the system is
85:32 coupled
85:33 to two heat bars
85:39 and these defaults are incompatible
85:41 there's one heat bath
85:43 that has a temperature zero and the
85:46 other bath that has a temperature
85:49 that is larger than zero that causes
85:52 really some fluctuations
85:55 now what are these heat buffs coupled
85:59 to now the final thing is that these
86:02 heat bars are coupled into time
86:05 so in one pound direction dt
86:08 smaller than zero yeah you have a zero
86:12 temperature
86:13 in the vicinity of the abdominal state
86:15 and in the other direction
86:17 d2 larger than zero never find that
86:20 temperature now and this two heat parts
86:23 coupled to different type directions
86:25 makes the system allow to go into the
86:28 absorbing state
86:30 but never leave it now that's this
86:32 asymmetry
86:34 that of these two incompatible bars
86:37 that makes this one of the hallmark
86:40 non-equilibrium systems in one
86:42 equilibrium
86:44 physics and what i showed you actually
86:46 here
86:47 this is extremely powerful
86:50 there's a lot of models that have
86:52 nothing to do with epidemics that fall
86:54 into this impossibility class
86:57 and so it's one of the
87:01 paradigmatic moments of non-acrylic
87:04 non-equilibrium statistical physics
87:09 okay so that was quite a tough lecture
87:12 yes
87:12 also for me i'm quite exhausted and what
87:15 i would
87:16 say is that uh you will have a great
87:19 christmas
87:20 and after uh the new year i'm joining
87:24 the fifth i think that's our next
87:27 lecture and then
87:28 actually we'll do something completely
87:29 different different and we have a look
87:31 at some real data
87:33 and we'll get into data science and see
87:35 what actually to do with data
87:38 this data is really really large how
87:40 actually you see these things that we've
87:41 studied
87:42 in the last lectures in the last three
87:45 months how to actually see that
87:47 in data now that's not very trivial if
87:50 somebody comes
87:51 up to you with 10 terabytes of data then
87:53 you can't just start matlab and start
87:55 phishing around
87:56 and you need special tools from data
87:58 science that allow you to extract
88:01 such features from data that can have
88:04 100 millions of dimensions that's what
88:07 we do right after the
88:09 after christmas on january 5th and when
88:12 once we've done that we'll also have
88:13 some guest
88:14 lectures done by real experts
88:18 in this field and
88:22 and once we've learned like the
88:24 fundamentals of data science
88:26 we'll have put that all together and
88:28 look into some actual
88:29 research data and see how we can
88:33 uh use these two tools from the
88:36 visualization
88:38 data science to actually dig into some
88:42 current experimental data okay so then
88:45 uh merry christmas everyone if you
88:47 celebrate that
88:49 and uh see you all next week next year
88:52 okay bye i'll stay there are
88:55 any questions
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