slide 1
00:00 there we go
00:07 okay so now you can see the screen i
00:09 hope
00:10 with a little overview so what where are
00:13 we actually
00:14 uh at the moment yeah so we had uh
00:18 two lectures ago we started thinking
00:20 about how order can emerge
00:22 you know and we said that this somehow
00:25 um very often uh relies on a balance
00:29 between
00:30 fluctuations you know that favor
00:31 disorder and
00:33 interactions that favor order
00:37 and then we went on next week last week
00:40 to study how we can have transitions
00:44 between different non-equilibrium states
00:48 for example between a
00:49 disordered and an order state or between
00:52 different kinds of ordered states
00:54 and we started that in a situation where
00:56 we neglected
00:58 moles we pretended that our
01:00 non-equilibrium system
01:01 or our general system could also be an
01:04 equilibrium system is very very large
01:06 and then we were basically back in the
01:08 framework
01:09 of non-linear differential equations and
01:12 non-linear partial differential
01:14 equations
01:15 and to understand then what kind of
01:17 order we had
01:19 we looked at specific situations here
01:21 where this order
01:23 or these patterns arose continuously
01:26 or not abruptly but they first were
01:28 small and then we can larger larger
01:31 and in these situations were allowed and
01:33 to linearize
01:34 and to ignore these difficult
01:38 non-linearities in these equations
01:42 so today we're now in a state where we
01:44 want to
01:46 join these two lectures now we're
01:49 looking at transitions between
01:51 non-equilibrium
01:53 steady states where
01:57 we have noise yeah and i wouldn't tell
02:01 you that i wouldn't have a lecture on
02:02 this if
02:03 the noise these fluctuations in these
02:06 transitions weren't super important
02:09 and maybe yeah then
02:12 once we've understood that we are in a
02:15 position
02:16 uh just to understand how does actually
02:19 order look like
02:20 how can we identify order and after that
02:23 we'll then
02:23 go to data and actually learn some tools
02:26 from data science
02:27 on how to extract such order
02:31 from complex and big data sets
02:34 and then at the end of this lecture of
02:37 this uh this term
02:38 uh we'll have a specific example from
02:40 research where we bring that all
02:42 together and we see
02:43 how this works together uh in the
02:45 context of
02:46 current research
slide 2
02:50 okay so to start
02:54 let's remind ourselves about
02:57 how we actually transit from order to
03:00 from disorder to order in equilibrium
03:03 yeah and we'll continue we'll
03:06 continue looking at continuous phase
03:09 transition and
03:10 in equilibrium these continuous phase
03:12 positions are characterized
03:14 by a critical point and this is just the
03:16 point where this balance
03:18 between fluctuations and interactions
03:21 this ordering and these disordering
03:24 forces
03:25 are just equal now the system doesn't
03:27 really know
03:28 exactly where to go whether to create an
03:30 ordered state or to a completely
03:32 disintegrated
03:33 order state and it's somewhere in
03:34 between and
03:36 uh so i i suppose
03:39 uh you've you you'll have had that in
03:42 your
03:42 statistical physics lecture normally
03:44 yeah but at the end of the first
03:46 statistical physics lectures but i'll
03:49 just give you
03:51 a little reminder of the most important
03:54 concepts
03:55 so here you see like this example from
03:58 equilibrium
03:59 this very powerful and intuitive model
04:02 system which is called the izing model
04:04 which is essentially modeling a
04:05 ferromagnet magnet
04:07 and on the left hand side you can see
04:10 simulations
04:12 for three different temperatures it's
04:15 just
04:16 such an icing model and what you see
04:18 here if the temperature is very low
04:21 you get an ordered state now everything
04:23 is black
04:24 all spins are putting the same
04:26 directions
04:27 in the same direction and if the
04:29 temperature is high
04:32 then you get a disordered state and this
04:35 disorder state as you see as a kind of
04:37 salt and pepper state now the average
04:40 magnetization is zero
04:43 but and locally uh you will always find
04:46 spin that goes up and down
04:48 go up and down and then you have that
04:50 thing in between these states
04:52 yeah when the temperature is exactly
04:56 equal to a critical temperature and this
04:59 is where
04:59 there is a balance between energy and
05:02 entropy of
05:03 fluctuations and order and
05:06 what you see here is this state
05:09 where you have these domains here
05:11 domains of these black domains
05:14 and if you zoom in to such a system
05:18 what you'll see at the critical point
05:19 you'll see that it looks exactly the
05:21 same
05:21 now you zoom in and you would wouldn't
05:24 be able to say whether this is a
05:25 snapshot a zoomed in version of the one
05:27 on the left hand side
05:29 or whether this is an entirely different
05:32 simulation
05:34 and then you can zoom in further and
05:35 further and
05:37 you again all the time get the same
05:39 impression that you can't really make
05:41 out
05:42 uh what is now the typical length how
05:44 large are these classes
05:46 you have here these domains of all sizes
05:49 equally now because you have domains
05:53 these white things or black things of
05:55 all
05:56 size is equally represented you can't
05:59 make out a single sound you can't make a
06:01 typical
06:02 length scale here because you can't say
06:04 that
06:06 the typical size of such a cluster here
06:10 is for example that large
06:13 as you always find clusters that are
06:15 much smaller you find glasses that are
06:17 much
06:17 larger here's a cluster now that has the
06:20 size of the entire system that's
06:22 infinitely large
06:23 and then you have all a spectrum of
06:26 sizes in between that
06:28 yeah well here in this example you kind
06:31 of get an idea
06:33 that these clusters are typically very
06:35 small
06:36 these domains whereas pin points up or
06:38 so are very small they have a
06:40 typical size but you can't see it on the
06:42 left hand side
06:44 yeah same as if i would zoom in with
06:46 this camera you know so
06:49 if we do like this now you immediately
06:52 see
06:53 that i zoomed in now because i have a
06:56 characteristic size i'm
06:57 one meter 80 or something yeah and now
07:00 you
07:00 see that you that have zoomed in the
07:02 camera and the picture is not the same
07:04 as before
07:06 so i'm not in a critical state yeah but
07:09 the icing system is in a critical state
07:12 and this critical state is characterized
07:15 by self-similarity
07:17 so they have structures of all sizes
07:19 represented
07:21 in this system now this is the
07:23 self-similarity
07:24 and the mathematical representation of
07:27 the self-similarity
07:28 the fact that you don't have an average
07:30 cluster size
07:32 is that the correlation length diverges
07:35 now so the correlation length is
07:37 infinity
07:38 that means the correlation function you
07:41 know
07:41 so how that represents how large
07:45 these clusters are is scale in variance
07:48 that means
07:49 that really classes of different sizes
07:51 are equally represented
07:53 and such if you ask what is the
07:55 probability that i
07:56 am now in a white cluster that i'm in a
07:59 black cluster a certain distance away
08:01 then the answer to this doesn't depend
08:03 on any specific distance
08:05 you know it's the same this probability
08:07 is the same this
08:08 correlation function here it's the same
08:11 when we
08:12 calculated in the original version of
08:15 the simulation or a zoomed in
08:17 uh fraction of this this is the
08:19 self-similarity
08:21 the self-similarity at a critical point
08:23 goes along with power laws
08:26 you know so the power laws have the um
08:30 for example like this here the critical
08:31 length as you go the critical
08:34 correlation length as you go closer to
08:37 the
08:37 critical point diverges to infinity it
08:40 goes to infinity
08:42 and it does so with an exponent that's
08:44 to be
08:45 called new when this gets zero here
08:49 this term will be infinity and these
08:51 exponents
08:52 capture how fast you go to infinity
08:56 and these exponents are very the fact
08:58 that you have an exponent but
08:59 that you have such a power rule means
09:02 that yourself similar you have a power
09:04 law if you have
09:06 something like this you can zoom in and
09:08 you still have the same exponent here
09:10 and you can't do that with an
09:11 exponential function or so
09:14 now and it also tells you that this is
09:17 here
09:18 uh if you have power laws from some
09:20 distribution that goes over
09:22 that has a power law that has this long
09:25 tail some exponent you cannot typically
09:29 calculate averages or moments because
09:31 these integrals diverge
09:34 now you have the power law of the
09:35 correlation function and you have a
09:38 power law
09:38 now if you have a power of the
09:39 correlation function you also have the
09:41 power laws
09:42 for example in the density and the
09:44 magnetization
09:46 uh near the critical point and all kinds
09:48 of other thermodynamic
09:50 quantities and i just wanted to briefly
slide 3
09:53 show you why this is the case
09:55 now it's actually the background of this
09:58 the background of this is actually an
09:59 assumption
10:01 that you say you have a free energy
10:05 and with this free energy uh
10:08 this free energy as you go to the
10:10 critical pound point uh
10:11 gets in finite it has singularity
10:15 and then you say that you assume
10:19 that the free energy here
10:23 this free energy has
10:26 one part that has all the physics and
10:29 all the details a regular part
10:32 but you say that the part of the free
10:33 energy that
10:35 diverges at a critical point
10:39 this one here so now it's is a
10:44 t t
10:46 is something like t minus
10:50 t c over t you know and h
10:53 is the external field if you have
10:56 something like this as a free energy as
10:58 the function of these parameters
11:00 yeah then the singular part the one that
11:03 goes to infinity
11:05 is a homogeneous function
11:08 homogeneous function just uh tells you
11:12 that if you have f of
11:15 lambda x that this is equal to
11:18 lambda to the power of some alpha
11:22 f on x yeah and this represents that
11:25 just this these gains zooming in you
11:28 have the same function you zoom in where
11:29 you rescale your variable
11:31 you zoom in and you get the same
11:33 function back
11:34 now this is a homogeneous function and
11:37 you still assume that this free energy
11:39 density in this case is a homogeneous
11:42 function
11:43 and this homogeneous function can only
11:45 depend
11:46 on dimensionless quantities now for
11:49 example
11:50 you wouldn't expect this divergence to
11:53 infinity
11:54 to depend on how you measure a length
11:58 now whether you measure the length in
12:01 units of centimeters or meters
12:04 whether you measure temperature in
12:07 kelvin or in units of one kelvin or two
12:10 kelvins or so
12:12 so you can see you these kind of things
12:15 these units
12:15 dimensions should be irrelevant for how
12:18 this quantity goes to infinity
12:22 yeah and if we say that then we say okay
12:24 so we have
12:26 here so-called scaling function that
12:29 depends on dimensional parameter
12:31 a dimensional dimensionless
12:34 combinations of our parameters so the
12:37 external field
12:38 divided to the temperature and then we
12:40 have to
12:41 take the temperature to some power of
12:43 something
12:45 to make everything dimensionless
12:48 so that it has no units no and there's
12:51 something that's not
12:52 something we don't know yeah and
12:56 this has the free energy has some units
12:59 therefore the whole thing gets doesn't
13:01 have the units you need a pre-factor
13:04 that gives you the right units you know
13:06 and then
13:08 you have this alpha which we don't know
13:11 yeah
13:11 this is some exponent that depends on
13:13 the specific model
13:14 for example for the eisenmann zero
13:17 you know and then you have these uh this
13:20 is the
13:21 consequence of how you translate this
13:23 homogeneity
13:25 here of the free energy to something
13:28 that you give names
13:29 as you have here this part that diverges
13:32 yeah that has the units
13:34 and this part here is the so-called
13:37 scaling function
13:38 that only depends on dimensionless
13:40 parameters
13:41 and it turns out that these exponents
13:43 and this scaling function are universal
13:45 so if you know it for one model then you
13:47 know it will know it
13:49 you will know it for a very large class
13:52 and we'll see that once we do
13:53 renormalization later today
13:57 so uh so what does it mean yeah so if we
13:59 make this
14:00 assumption that's really an assumption
14:02 about
14:04 homogeneity of the free energy then we
14:07 can calculate for example
14:09 the magnetization m of th
14:12 now this is in thermodynamics something
14:14 like
14:16 del f 2 del h
14:19 you know and then we just plug this in
14:22 and we get something like temperature
14:25 this reduced temperature
14:27 t to the power of minus 2 minus alpha
14:30 means
14:30 minus delta some other function that we
14:34 don't know
14:35 that again depends on a dimensionless
14:40 parameter and then
14:44 this scales like some better that's the
14:47 definition
14:48 of this exponent better of the
14:50 magnetization
14:52 yeah and uh so in thermodynamics this is
14:55 called
14:56 rhythm scaling basically in any textbook
14:59 on statistical physics
15:00 and it's just just to show you how the
15:04 assumption
15:05 of homogeneity near the critical point
15:09 leads to power laws in other
15:11 thermodynamic quantities
15:13 i've shown you here the magnetization
15:17 this was the magnetization
15:23 but the same holds true for example for
15:26 susceptibility
15:27 and other thermodynamic quantities that
15:29 you can get by taking derivatives of
15:31 your
15:32 energy so
15:35 this homogeneity or this self-similarity
15:39 that i showed you here that is a
15:41 reflection that is one of the hallmarks
15:44 of uh critical behavior and that's what
15:47 we're looking for when we look for
15:49 critical behavior
15:50 and now the question is can we see
15:53 something like this
15:55 also a non-equilibrium system
slide 4
15:58 before before i start with that
16:01 uh let's just have a look at one
16:02 specific how this scaling
16:04 behaves if you look at these equations
16:08 here
16:10 what does it mean it means that
16:13 actually the curves that you get you
16:16 know so if you just divide so once you
16:19 make a measurement for example with a
16:21 known temperature
16:23 and a known magnetic field you measure
16:26 this function here
16:28 then you know that it doesn't depend
16:29 separately
16:31 on the age and the temperature
16:35 so you can rescale your axis so this is
16:37 what is your y
16:38 x axis you can use that your x axis and
16:41 your y
16:42 axis to make all of these curves
16:46 collapse onto each other yeah and this
16:49 is this uh
16:50 scaling form that we see in equilibrium
16:54 physics this is for the icing model
16:56 so on the x-axis we have this scaled
16:58 temperature
16:59 that would be t on the previous slide
17:02 lowercase t
17:04 uh times something so this is v scale
17:08 and then these uh people in these
17:11 experiments
17:12 for uh for
17:15 for a ferro ferromagnet measured
17:18 the magnetization for different values
17:20 of the
17:22 of different experimental parameter
17:24 values
17:25 like magnetic field external magnetic
17:27 field temperature
17:30 and by making use of this formula here
17:33 you see that uh the scaling
17:37 behavior what is where is my
17:40 is it going down here yeah that's the
17:42 scaling behavior here
17:45 yeah that the only thing
17:49 that you don't know is this g of m that
17:52 you have to measure
17:53 yeah once you know the h and the
17:56 temperature
17:58 you can make all of these different g of
18:00 m's the gms
18:02 this scaling function you can rescale
18:05 these axes
18:06 to make them collapse onto each other
18:09 yeah and this is this observation this
18:11 is how you observe
18:12 scaling an experiment so you manage to
18:15 collapse
18:16 your experimental curves by multiplying
18:19 this x-axis and the y-axis with certain
18:23 values
18:23 of offense you have to guess you can
18:27 collapse all of these curves on the same
18:30 uh universal so-called scaling form
18:34 now this is the manifestation of scaling
18:36 and that's of course
18:37 also something we'll be looking at and
18:39 non-equilibrium systems
18:41 but also in data
18:44 now scaling is a whole mark of critical
18:48 behavior and today
slide 5
18:52 we want to see whether these concepts
18:56 of scaling and criticality yeah
18:59 where and these these continuous
19:02 phase transitions actually also extend
19:05 to non-equilibrium systems
19:07 yeah and it turns out so now we first we
19:10 need to find a non-equilibrium system
19:12 that is as intuitive
19:15 as the ising model and the icing model
19:18 is very intuitive
19:19 you have that in your lectures when
19:21 you're a student and
19:22 in your later life as a scientist you
19:24 always refer to that because it's so
19:26 simple and intuitive that uh you can
19:28 explain a lot of things a lot of
19:30 things about continuous phase
19:32 transitions in equilibrium
19:33 just based on this very simple model
19:35 like i did in the beginning of this
19:37 lecture
19:38 and it turns out now that the uh
19:41 icing model of non-equilibrium physics
19:44 of course is
19:46 that so people would dig a disagree but
19:48 one of the simplest models in
19:49 mono-equilibrium physics that shows
19:52 critical behavior
19:53 is an epidemic model and this epidemic
19:56 model we knew already
19:57 from the previous lectures here we have
20:02 our good old si model again
20:06 now so this epidemic model is this is
20:08 the simplest model
20:10 that you can think about so you have
20:11 infected individuals
20:14 i and susceptible individuals or
20:17 healthier people's less
20:19 now if an i meets an s
20:22 then the s gets infected with the rate
20:25 say lambda half
20:27 and turns into an affected infidel and
20:30 in the end you have two of them
20:33 then you have the other process that we
20:34 recover
20:36 and we set this rate to one now we can
20:38 just set that to one
20:40 without any loss of generality and uh
20:43 so that infected we measure units
20:46 time in units of this recovery rate
20:50 you know service infected individuals
20:52 can also
20:54 then recover and become
20:57 healthy again we have these two kinds of
21:00 individuals
21:01 and now we put them in the real world so
21:04 last time we were only looking at some
21:06 well-mixed
21:07 average quantities but now we put them
21:10 into the real world like the city of
21:11 brisbane also
21:13 where they actually can where actually
21:16 space
21:17 matters yeah so i'm more likely to
21:19 infect somebody else working at the
21:22 mp rpk pks than somebody looking at
21:25 another max planck institute for example
21:28 yeah so
21:29 so here uh these spatial structures
21:32 these special degrees of freedom
21:34 uh are taken into account and the
21:36 simplest way of you
21:38 how you can think about this is at the
21:40 bottom here
21:42 now that you look at letters
21:45 you have a letters each site
21:48 carries either an affected individual or
21:51 a recovered individual and
21:55 you know an infected or a recovered
21:56 individual and
21:58 when an infected individual
22:02 is next to a recovered a healthy one
22:05 then
22:05 the healthy one can turn into an
22:07 infected one
22:09 with a certain probability of with a
22:10 certain rate lambda over two
22:13 yeah and also there's another process
22:16 here if i saw if the
22:18 individual on a certain position is
22:21 infected
22:21 it can turn into a healthy one at a rate
22:24 lambda
22:26 so this is this simple spatial version
22:29 that you can think about for this and
22:32 simple epidemic model and it's also the
22:35 literature is often called the contact
22:38 process
22:40 so of course real epidemic model models
22:43 have
22:44 typically one more component namely the
22:47 um
22:49 [Music]
22:51 the infected recovered uh wait
22:54 is this also here okay so so the third
22:57 component that you normally have
22:59 and these models are the recovered one
23:01 the immune people
23:02 now you have the disease yeah and then
23:04 you are fine for the rest of your life
23:06 and you're immune to this disease
23:08 so you can only forget in fact one then
23:11 you have a third
23:12 species here a third kinds of particles
23:15 which would be the recovered ones
23:17 or the immune ones and they cannot be a
23:21 faculty again
23:22 but this slightly more complicated model
23:25 uh is shows very similar behavior to the
23:28 model that we're studying here
23:31 for the things that we're interested in
23:32 so here we're interested in infinities
23:34 in singularities so once you
23:38 once you look at these kind of things
23:40 then these models will qualitatively the
23:42 same
23:43 although also the exponents will be
23:45 different
23:47 but once you look of course into
23:49 non-singularities it's more critical
23:51 behavior
23:52 than the messiness of how wide your
23:55 roads are
23:57 how often the tram goes uh between the
24:00 blasphemy institutes and so on these
24:02 things will matter
24:05 yeah but close to the critical point uh
24:07 we'll be fine
slide 6
24:10 so this is a stochastic simulation of
24:13 such a system
24:14 and we can just see what happens on the
24:17 left hand side
24:18 you see a simulation of such a lattice
24:20 system
24:21 where you initially have random random
24:25 random initialization so every site
24:28 is either with the probability of one
24:30 half
24:32 a certain probability infected or
24:35 not infected and what you see here
24:39 now is in blue infected
24:42 individuals now if this lambda
24:46 now this infection rate is smaller
24:49 than a certain critical value
24:52 then what you will see is that this
24:54 infraction
24:55 this infection can spread for a while
24:58 but most of the time with a certain
24:59 probability
25:00 it will uh disappear
25:04 now so in this regime here in this phase
25:08 the recovery rate outweighs
25:11 the infection rate you know and
25:14 uh so that's what we're supposed to be
25:17 on in this regime we're supposed to be
25:19 investing
25:19 starting next week and then on the right
25:23 hand side
25:24 that's the regime that we're currently
25:26 in now then the infection rate
25:28 is larger than the uh
25:31 than the recovery rate now so the
25:33 infection probability is higher
25:36 and what you will then end up is is a
25:39 state where most of the individuals will
25:42 carry the disease will be infected
25:45 so you will you will reach a steady
25:47 state
25:49 not everybody is all the time in fact
25:50 that you will reach some steady state
25:52 with a certain percentage of infected
25:55 people
25:57 and now we have this situation in
26:00 between
26:02 that's this one here and this is
26:05 where the um where
26:08 the infection rate is more or less
26:11 balanced
26:12 with the recovery rate it's not exactly
26:15 equal to one
26:16 so that's these things are complicated
26:18 yeah you might think that
26:20 okay if this this lambda is equal to one
26:23 or one half or so
26:25 yeah then uh then that's the critical
26:27 point of these systems i'll show you are
26:29 more complicated than you might think
26:32 because the noise is so important here
26:35 and here
26:36 what you see is that you have domains
26:39 that become larger and larger over time
26:41 so from
26:42 when we go from top to bottom we have
26:43 time now so we go we start here at the
26:46 top
26:47 and then this domain goes large and
26:49 larger you have merging
26:51 of domains with infected individuals
26:54 and then we have branches that they die
26:56 out
26:57 like this one here and uh
27:00 it looks a little bit like a
27:02 self-similar state
27:04 now we have domains of all sizes for
27:07 example
27:08 in the time domain or from top to bottom
27:11 you have some branches that die out
27:13 quite quickly here
27:15 but then you have other branches like
27:17 the big one in the middle
27:19 where that just go on for uh forever
27:23 without really occupying the whole
27:24 system
27:26 and then if you look take a slice in
27:28 this direction here
27:30 these simulations are very small also
27:33 it's not like a design
27:34 you can't see that well you'll also see
27:36 that here you have
27:37 structures of all different sizes
27:41 there are small ones like this one here
27:44 yeah and the larger ones like this one
27:46 and so you have these structures of all
27:48 different sizes
27:51 and this is again reminiscent of cell
27:53 similarity
27:54 and the critical point so it turns out
27:57 our little empiric model has a critical
28:00 point
28:00 can i ask a question um i i think i
28:04 roughly understand the model
28:06 but this simulation is the simulation of
28:08 what
28:09 so is it a hamiltonian system where the
28:12 um just yes so what is this basically
28:15 nothing
28:16 yeah so uh you just take what it is here
28:19 or you take that just this year the way
28:21 here that you write these simulations
28:23 the different ways to write them you
28:25 have a lattice you know you have a
28:26 numerical simulation you have a vector
28:28 an array and you either have like
28:31 one or zero and then you pick a side
28:35 randomly
28:36 and perform these reactions here
28:39 you know so so the one way to do that is
28:41 to pick a side randomly
28:44 and check if your neighbors overcome if
28:45 you pick this side
28:47 and if your neighbor is susceptible or
28:49 it does not is not infected
28:52 then you infect the neighbor with the
28:53 probability
28:55 lambda over two so like a monte carlo
28:58 simulation it's a monte carlo simulation
29:00 the different ways you can also think of
29:02 a cellular automaton
29:04 yeah but uh the typical way to simulate
29:06 these things are multicolored
29:08 simulations
29:09 okay but there's no hamiltonian there's
29:10 no deeper insight you can just take
29:12 these rules
29:13 and simulate them on a lattice and the
29:15 only thing you have to do
29:16 is to take into account that this is not
29:19 a deterministic process here
29:21 but it's a random process with a
29:23 probability one-half
29:25 you turn this one here lambda over two
29:27 you turn this one here
29:29 into an infected blue one can i then
29:32 properly
29:33 um make a make a statement about the
29:36 time scale how
29:37 some how this thing spreads because it's
29:40 it's random right so i can
29:44 that's what we'll be trying to do today
29:47 okay uh but we'll
29:48 uh only be managing to do that tomorrow
29:50 uh let me just go on
29:52 so here you get some kind of idea
29:55 already
29:56 in this slide here you can get us some
29:58 kind of idea here so that
30:00 that you have here a time scale yeah
30:03 it's not
30:04 where things uh where things uh
30:06 disappear
30:08 yeah so you say that for example here
30:10 the typically the
30:12 the number of infected individuals will
30:14 go down with an exponential function
30:17 yeah and then this has a typical time
30:18 and then you get rid of most of the
30:20 infected ones
30:22 this has some certain time at this
30:25 typical
30:25 this certain time where you say okay
30:29 at this time i'm i have lost most of my
30:31 infected
30:32 people now that they're healthy again
30:35 this is then called
30:36 psi parallel
30:39 yeah so this is like a correlation
30:41 length so i'm actually actually getting
30:43 a hat a little bit too far so so this
30:45 you have here a correlation length in
30:47 time
30:48 but that tells you exactly that how f
30:51 how quickly
30:52 does this disease disappear
30:55 yes you have a correlation length in
30:57 space like this
30:59 so for example this one here
31:03 now our analysis is it depends on how
31:05 you define it it can relate to that
31:07 uh this is typically called psi
31:10 perpendicular
31:12 you have a correlation length in space
31:13 now that tells you how large are your
31:15 clusters
31:16 but you also have a correlation length
31:18 and time how long
31:20 lived are your clusters how long does it
31:23 take for them to disappear
31:25 and it turns out that both of these
31:27 things at the critical point are
31:29 infinite
31:30 so the system is not only self-similar
31:32 in space but also in time
31:37 yeah but first before we before we do
31:39 that um
31:40 uh before we do that formally so what i
31:43 see here
31:44 is is the stochastic simulation uh we'll
31:46 later
31:48 motivate some larger equation that we
31:50 actually will be studying
31:52 but for now now the system is as simple
31:54 as it gets now you have a neighbor
31:56 if this neighbor is not infected you
31:58 infect it with a certain probability
32:00 yeah it's the simplest there's like five
32:03 lines of code also
32:04 in matlab now there's nothing there's
32:08 nothing
32:09 uh in terms of the simulation the modal
32:11 definition is nothing
32:12 that is nothing deep in there but of
32:15 course the consequences as we see on the
32:16 slide
32:17 are rather non-trivial
slide 7
32:22 so now we want to go one step
32:26 ahead and try to formalize this
32:29 mathematically
32:32 and um to formalize this we first need
32:36 to
32:36 have something to put into our larger
32:39 equation
32:40 yeah and that something that we put into
32:42 our laundry equation
32:44 is the density of this or this order
32:47 parameter
32:49 is the density of infected
32:52 individuals all right to get this we uh
32:58 um we we do a double average
33:02 so this average here is over the lattice
33:06 now we sum over the letters and we count
33:10 now with this si variable like a spin
33:14 how many infected individuals we have
33:17 and divide it by the total number of
33:19 lattice sites
33:20 the system size and then we average
33:23 again
33:23 over the ensemble now this is our order
33:26 parameter and this parameter this order
33:28 parameter
33:29 tells us whether we have order or not
33:33 you know if this is one then everybody
33:35 is infected
33:36 yeah it's not or not order or not if
33:38 this is one
33:39 everybody's infected and if this is zero
33:42 then everybody is healthy
33:45 so this is our like our magnetization
33:48 and now
33:49 we want to do the same thing as an
33:52 equilibrium
33:53 i also want to ask what are we actually
33:56 looking at yeah so
34:00 what we say is we don't know
34:04 but we make the assumption
34:07 that in this non-equilibrium critical
34:10 point
34:11 we also have scaling behavior and we
34:14 also have self-similarity
34:16 now and of course you can test this
34:18 assumption if you do large enough
34:20 computer simulations
34:22 so one thing is that this
34:25 if our system obtains a steady state
34:28 with some density
34:30 you know so that's a row
34:33 stationary density something like the
34:36 magnetization you know the process of 90
34:39 of the people
34:40 are infected uh
34:43 this goes with
34:48 lambda minus lambda c to the power
34:51 of beta it's like the magnetization we
34:54 don't know what beta is
34:56 but there is some better that we want to
35:00 know
35:02 now as i've discussed already before we
35:04 have now not just
35:05 one correlation length but two so one is
35:08 the spatial
35:13 correlation length
35:18 and this is typically denoted by psi
35:21 perpendicular
35:22 because it's perpendicular to time and
35:26 perpendicular so if you look at these
35:28 pictures here
35:30 you can kind of get an idea
35:33 why this is called perpendicular and
35:35 parallel
35:37 suppose that this is here whether this
35:39 actually an
35:40 equivalent model is the one of water
35:43 pouring
35:44 into soil you know so you have little
35:48 channels
35:48 it's a rough thing yeah and then for
35:51 example here
35:52 the water flows down
35:56 but at some point now the density of the
35:59 soil is too large
36:00 and the water stops this is
36:03 this is an example where the soil is
36:05 like the soil is like
36:06 it's very open it's not very dense you
36:09 put water in it
36:10 and it flows all the way to the bottom
36:13 so that's
36:14 what is what is sorry yes um you said
36:17 that
36:19 in case of critical systems the
36:20 correlation length in space can be
36:22 divergent
36:24 yes and also the correlation length in
36:27 time could be divergent
36:28 yes so if the correlation length in time
36:31 is divergent then in this specific
36:33 example
36:35 the it'll the number of infected
36:38 clusters will always be present
36:40 right yes exactly you will not you will
36:43 never get rid of this
36:44 disease but of course this infinities
36:47 when i talk about infinity
36:49 these infinities are not defined really
36:52 in this small simulation where we maybe
36:54 have
36:54 100 individuals also now these
36:57 infinities are defined for
36:58 systems that don't really have an
37:00 infinite infinitely large size
37:03 now this here what you see the
37:05 simulation in the middle
37:07 can just by chance disappear
37:11 and it will disappear and i can tell you
37:15 even that the disease in this case here
37:18 the right hand side will disappear with
37:21 a very small probability
37:23 right so it's a very nice feature of
37:25 this model that will turn out to be very
37:28 important
37:29 what happens if all individuals
37:32 are healthy what happens if all
37:36 individuals are healthy
37:39 then there's no process here
37:44 there are only s's there's no process
37:46 here that can give you the disease back
37:49 once the disease is extinct
37:52 it will never come back and because this
37:55 is a stochastic system
37:57 you just have to wait long enough and
37:59 just by chance
38:01 even this is casey on the right hand
38:03 side will turn into the
38:05 case just because it's stochastic just
38:08 by chance
38:09 maybe you have to wait 100 billion years
38:11 or so for this to happen but you know
38:13 that at some point you will end up in
38:15 this state
38:17 where the disease went extinct by chance
38:20 you have to wait extremely long for that
38:22 but you know that it will happen
38:24 and these states here now like in this
38:27 system here
38:28 you go to zero and then there's no way
38:30 it can come back
38:32 yeah in reality you will have to wait
38:34 for evolution
38:36 to create another virus that has the
38:39 same properties
38:40 now to come back so that texas goes
38:42 extremely long
38:43 yeah it's a much it's much longer than
38:45 the spreading of the disease itself it
38:47 happens in one or two years
38:50 you know and so these are called
38:52 absorbing states you can go in there
38:54 but you can never go out again
38:58 so in other words this means that in
39:00 these absorbing states
39:02 they're very important for not only for
39:03 virus spreading but in any ecological
39:05 model
39:06 now we have extinction and this
39:09 absorbing states
39:10 uh you can get in but you will never be
39:12 able to get out
39:14 now once you're in there you're trapped
39:16 and these absorbing states they don't
39:18 have
39:19 fluctuations they don't have any noise
39:21 and we'll see that in the larger
39:22 equation you know so these absorbing
39:25 states don't have any noise
39:28 and by this you can already see that
39:30 this whole system
39:31 is a non-equilibrium system because if
39:34 you have a state that has no noise
39:36 this is not a thermal system where you
39:38 have a temperature
39:40 now so this here is a system where you
39:41 have noise now for example here you have
39:44 noise but once you reach the state
39:47 where there's no disease no virus left
39:51 you don't have any noise anymore you
39:54 know and that cannot happen in a
39:55 thermodynamic system that is an
39:57 equilibrium
39:58 that you always have your temperature
40:00 and this will always give you noise
40:01 regardless of how many
40:03 particles you have or whatever yeah so
40:06 this already tells you that this is a
40:07 non-equilibrium system
40:09 and it's a very interesting system and
40:11 the system is actually one of the
40:13 universality classes non-equilibrium
40:15 physics
40:16 so that's once you are getting a little
40:19 bit ahead
40:20 once you know that your system has one
40:22 absorbing state
40:24 many ecological systems for example one
40:26 absorbing state
40:28 then it's quite likely that what i'll
40:29 show you in these
40:31 uh renewalization calculations today and
40:34 next week
40:35 will also apply to these systems is a
40:37 very powerful
40:38 you know universality class and
40:40 universal system
40:42 for non-equilibrium systems
40:45 yeah but let's let's i was here talking
40:47 about did i actually answer your
40:49 question so i got a little bit
40:50 uh distracted uh i i distracted myself
40:55 a little bit yeah did i do that
40:58 okay okay i i forgot it at the end i
41:01 forgot this question but i hope i
41:02 answered it at some point
41:04 okay so you have the two correlations
41:06 just spatial correlation length
41:07 you know that's uh sigma
41:11 uh side perpendicular and actually
41:14 what i want to say here is actually
41:16 that's that's what i would say
41:18 yeah so you have here you have soul and
41:20 you have water
41:21 flowing through this then the parallel
41:23 length here
41:26 this one it's called parallel because
41:28 it's parallel to the direction of
41:29 gravitation
41:31 and the other length here is
41:34 perpendicular because it's perpendicular
41:36 to the
41:37 direction of gravitation now that's
41:38 where these names come from
41:40 because these models called direct or
41:42 the directed percolation
41:44 is that you have something flowing
41:47 through a rough
41:48 medium like soil and then you have a
41:51 direct gravitation force that pulls the
41:54 fluid into one direction
41:55 but not in the other direction and
41:57 that's where these parallel and
41:58 perpendicular
42:00 yeah so and then we give that some
42:02 exponent
42:04 lambda minus lambda c to the power of
42:08 minus
42:09 mu perpendicular
42:12 now that we have the temporal or dynamic
42:22 correlation length
42:25 side parallel and this
42:28 we call and very surprisingly minus
42:32 new parallel
42:35 and this is now as you said so
42:38 our temporal correlation can become
42:41 infinity
42:42 what does it mean yeah so i have a
42:45 perturbation
42:46 to the system so what so if you have a
42:49 the spatial correlation and infinity
42:52 like an isomorph
42:53 you make a perturbation and this
42:56 perturbation will in principle
42:58 affect all parts of the magnet
43:01 yeah you will have a very very long
43:03 range correlation you flip a spin
43:05 somewhere
43:05 and it has an effect somewhere
43:07 completely somewhere else
43:10 now we have an infinite correlation
43:13 length
43:13 in time what does that mean so that
43:16 means that if we perturb the system we
43:18 are at a critical point
43:19 we will perturb the system and the time
43:23 that it takes the system to go back to
43:26 forget this perturbation
43:27 is infinitely long yeah
43:30 so so you have again now processes in
43:33 all time scales and parallel
43:35 very long very slow process and also
43:37 infinitely long processes
43:39 yeah it's like like the space in the
43:41 isis mode you have classes of all
43:43 different sizes now you have also
43:44 processes
43:45 of all different length scales at the
43:48 same time
43:49 now and this is what this criticality
43:51 does to time
43:52 the time domain you make it perturbation
43:55 and it never just disappears again
43:57 that the effects of this particular
43:59 perturbation you will see in this system
44:01 infinitely long now so that's the cool
44:04 thing about critical systems that does
44:06 uh
44:08 it does uh they do very straight things
44:11 and then now we define another uh
44:15 i've got a question yes sorry is are we
44:18 still dealing with a mean field model
44:21 uh i i didn't tell you yet uh but we'll
44:24 we won't be dealing with the mean field
44:26 of model
44:27 yeah so mean field is not very good for
44:30 these kind of things
44:32 yeah so we're not uh we're not dealing
44:34 with the mean fit model
44:35 last last time last week we were dealing
44:37 with mean field models
44:39 but this time we have to take propaganda
44:41 fluctuations properly into account
44:44 and we will have also to take into
44:46 confluctuations on all
44:48 different temporal and spatial scales
44:51 you know so that's that's what we will
44:53 have to do and that's what we will uh do
44:55 with the renovation group
44:58 so mean field theory is typically pretty
45:00 bad for these things
45:03 even just for the getting what is this
45:05 lambda c i'm going to show you what this
45:06 lambda c is but this is
45:08 in the mean field version we would say
45:10 okay this is just one half or something
45:12 like this
45:12 yeah where you write down some
45:14 differential equation like you did last
45:16 time
45:16 you write you guess some differential
45:18 equation you motivate it
45:20 and then you get some lambda c but i'll
45:22 show you today
45:23 now that this is actually not how it
45:26 works if you have these
45:27 strong fluctuations to get different
45:29 results
45:31 okay so then we have a third exponent
45:35 the dynamic critical exponents
45:39 and that means that near lambda c near
45:42 the critical point
45:45 we say that uh these correlation lengths
45:49 now they they both follow power rules
45:52 you know and we say that they are
45:54 connected
45:55 by this exponent called z
45:59 and this is called
46:02 a dynamical
46:07 critical
46:11 exponent yeah so we what we did now
46:14 is okay we said okay these systems look
46:17 self-similar
46:18 uh like in equilibrium and we assume
46:21 that the same concepts of scaling
46:23 and power laws also apply to
46:26 non-equilibrium systems
slide 8
46:30 and now i have a slide here that we
46:31 already talked about
46:33 this is just what are these correlation
46:35 lengths now so what are these
46:37 correlation uh
46:40 what are what are these correlation
46:41 lengths these two and we discussed that
46:44 basically already
46:45 and so if you look here on the right
46:49 hand side for example
46:51 so on the left hand side i have two
46:53 simulations
46:55 that were started from an initial seat
46:58 from this just one
46:59 infected individual and on the right
47:02 hand side
47:03 these figures uh they are from
47:06 assimilation
47:06 were maybe 50 percent of the letters
47:10 was infected you know other
47:13 simulations i didn't show you uh there's
47:16 a nice review
47:17 article by hindi uh
47:24 about a non-equilibrium criticality and
47:26 face transitions
47:28 and that's where it took this pair this
47:30 uh it's a very nice review
47:32 uh about the kind of things that we're
47:34 doing this week and next week
47:37 so here now we this is just an intuition
47:40 about what these correlation lengths are
47:42 uh i told you already you know that this
47:45 um that this temporal correlation length
47:49 side parallel gives you to say the time
47:52 that the disease
47:54 dies out as you can see that here
47:57 in this simulation here that the
47:59 parallel correlation length
48:01 that's the time for such a droplet here
48:04 to go away again now we know that
48:08 if we're below the critical point that
48:09 at some point it will disappear
48:12 and this has a typical time and this is
48:14 just the excite
48:16 parallel and then you have also a
48:18 typical size of such a droplet here
48:21 that's psi perpendicular that's just how
48:24 large
48:25 do these domains get and you can also
48:28 see that here on the right hand side of
48:29 course
48:30 how long does how large does the domain
48:33 of uninfected
48:34 or infected individuals get now there's
48:37 this one here and how long does it
48:39 survive
48:41 now that's how these correlation lengths
48:44 are to be
48:45 interpreted intuitively
slide 9
48:49 okay now
48:53 i'm going to do a little step
48:57 uh that where i'm trying to avoid a long
49:02 calculation
49:05 what we usually would do now is we would
49:08 take this lattice model and we would
49:11 try to derive a master equation
49:15 and also this probability that the
49:17 lattice
49:18 has a certain configuration and then we
49:21 would be looking
49:22 at this master equation uh
49:25 this very complex master equation that
49:27 tells us the time evolution of this
49:29 vector and we try to get the rates
49:33 and then we would uh do approximations
49:36 uh like the system size expansion or the
49:38 chromosomes
49:39 expansion and so on and then we would
49:41 try to derive this large
49:43 which now that's a lengthy business and
49:46 it's not the subject of our
49:48 lecture what i just want to show you is
49:52 that why does this larger equation that
49:55 i'll show you
49:56 later not look like what you naively
49:59 would expect
50:01 to this end you can just show you that
50:03 such a letter
50:04 system you can interpret in different
50:06 ways
50:08 one way is to say that it won't give you
50:11 like
50:12 the mathematical rigorous form of the
50:14 larger equation but it shows you why
50:16 it looks not like you expect it to look
50:19 like
50:20 so the first way we can interpret
50:24 this lattice system is to say what
50:27 happens
50:28 at t and what is the state at some d
50:32 plus dt some like a very short time
50:35 interval after that
50:38 and now i'm taking like in this master
50:40 equation i'm taking the perspective
50:43 of the state in a certain site
50:48 and then ask what are the previous
50:51 states
50:52 that give rise to me being infected
50:57 you know what gives rises yeah and
51:00 suppose that i was not infected before
51:03 yeah then my left neighbor could have
51:05 been infected
51:07 my right neighbor could have been
51:09 infected or both of them could have been
51:12 infected
51:12 and have affected me these are the three
51:15 processes that
51:16 lead to me being infected and then
51:19 if it was there's another process
51:23 and i should use here different colors
51:25 it's not the recovery
51:35 i switched the colors not red is
51:39 healthy sorry
51:43 red is healthy and blue is infected
51:48 okay yeah if i in this recovery process
51:53 it doesn't matter what my neighbors were
51:54 i would just i just know that was
51:56 previously infected
51:57 if i'm now in the process of recovery
52:01 now this is this master equation picture
52:03 where we ask the word which state do i
52:04 come from
52:06 and now we can take an equivalent
52:09 description
52:10 and take more a lattice picture now that
52:13 corresponds more a little bit left to
52:14 the longer
52:15 picture yeah where update where i say
52:18 what is the state of the lattice now
52:20 and what is the state of the lattice the
52:21 next step
52:23 now that corresponds to this
52:24 differential equation picture
52:28 yeah and then how can i update it so
52:32 then i need at least two lattice points
52:35 at the same time to update to define
52:37 these updating rules
52:39 and then i have different possibilities
52:41 here
52:42 yeah if my two lattice points are
52:45 infected
52:46 that previously either the left one or
52:49 the right one
52:50 was infected
52:54 the recovery process is no more
52:56 complicated
52:57 yeah if i have known that in this two
53:00 side picture i have one
53:03 infected and one one infected and one
53:07 not affected once this white is down
53:09 here
53:10 this is
53:13 healthy this is
53:17 infected
53:21 now if i have one infected and one
53:24 healthy one
53:25 previously my system could have been
53:29 must have been in a state now if i'm
53:31 looking at the
53:32 recovery process where both of them were
53:35 infected
53:37 now so with the probability one half i
53:39 have either this
53:41 or this one here and then
53:44 if both of the states in the second time
53:47 step
53:48 are healthy then one of them was in fact
53:52 before now so here i update two sides at
53:55 the same time
53:57 or i can also say i updated the whole
53:59 letters at the same time
54:00 in parallel and now
54:04 the thing is what is this here
54:09 what is this here these two processes
54:11 here
54:12 suddenly i have a process a process for
54:15 recovery
54:17 that involves two individuals
54:20 yeah formally that involves two
54:22 individuals so here have two individuals
54:25 to infect it
54:26 and after that only one of them is
54:28 infected
slide 10
54:30 now suddenly i have two individuals and
54:32 if i ask how
54:33 such a term here if i take this
54:37 description as the basis for my larger
54:40 equation
54:41 how will this term pop up in my larger
54:44 equation
54:45 then it's this term is proportional to
54:49 one-half
54:50 times
54:55 the probability that one of them is
54:57 infected and the probability that the
54:59 other one is
55:00 also infected so we will have something
55:03 like rho
55:04 of x t squared
55:08 and we'll get a minus because we
55:10 decrease the number
55:11 of infected individuals
55:15 now this is just to show you yeah by
55:16 this uh
55:18 magnesium inside the exotic lattice
55:20 representation
55:22 that you suddenly get a term you write
55:25 down the larger equation here
55:27 this is the larger equation
55:30 and this is here what you expect the
55:32 first thing is what you expect is an
55:34 infection term
55:35 you have the density of infected people
55:37 at a certain position x
55:40 and this depends on how many infected
55:43 people i already have
55:44 that is this typical exponential
55:47 increase of the infection rate
55:51 now we get another term here
55:55 this term that describes the recovery
55:58 process now this describes
56:02 the recovery process and
56:05 it suddenly has this quadratic term
56:08 although this recovery process like this
56:10 picture looked completely linear because
56:12 every individual was doing it
56:14 individually now we have now the second
56:18 degree term here and that looks like an
56:21 interaction
56:22 and this comes just because we're
56:23 updating all the letters in parallel in
56:25 this
56:26 lingerie equation and that's why we get
56:29 this
56:30 second degree term here we have a
56:33 diffusion term
56:34 this one here that was also not in our
56:36 model description
56:38 you know that we never said that these
56:41 particles are actually moving around
56:44 but there is some spread of spatial
56:46 information because you
56:47 interact with the nearest neighbor yeah
56:50 and this
56:51 is models if you zoom out and go to a
56:54 continuous picture
56:56 it's a spread diffusive spread of
56:58 infection information
57:00 and that's why you effectively get a
57:02 term here you need to have a term
57:05 that involves spatial derivatives now
57:07 where you actually spread something over
57:09 space you spread the infection of
57:11 space and that's why you get this term
57:13 here
57:14 and you have of course again a noise
57:16 term
57:18 now and i'll give these here parameters
57:20 and also new names the combinations of
57:22 the old parameters
57:24 and we want to keep this we want to have
57:26 different parameters at each of these
57:28 rates because we in the next step want
57:30 to renormalize these parameters that we
57:33 we need them and the noise here
57:36 on the right hand side is our good old
57:39 gaussian noise now that has
57:41 zero mean and
57:44 correlations in space and time
57:47 that are luckily delta distributors so
57:50 they're
57:50 memory less they don't have memory in
57:52 space or in time
57:54 but they depend on this density here
58:00 they depend on this density and what
58:03 this means
58:04 is that this noise
58:08 the correlation
58:11 in the noise or the strength of this
58:13 noise that's the strength of
58:15 noise
58:19 is zero
58:24 if the disease
58:28 is extinct that's called
58:31 multiplicative noise because now the
58:33 density
58:35 rho of x and t is a pre-factor in the
58:39 noise term that
58:40 if is it is contributing
58:43 or defining the strength of the nodes
58:45 and once we have
58:46 zero infected individuals left then the
58:49 noise disappeared and we can never have
58:51 get away out of this term out of this
58:53 point
58:54 where the infection is lost
slide 11
58:59 so maybe
59:02 quite late in time as a next step
59:10 um we get a little bit more formal so
59:12 now we have the larger
59:13 equation now and what you see here
59:16 already
59:17 is the martensite rose functional
59:20 integral that we derived
59:22 and this functional martin citra rose
59:24 function integral
59:25 or martensite rules johnson the dominic
59:27 is functional integral
59:29 you can divide very easily we remember
59:32 the equation that we had
59:33 a few lectures ago first
59:36 part of this function integral
59:42 what was it here
59:45 it's just the launch of a equation
59:48 yeah that that makes sure that you
59:50 actually solve the launcher equation
59:52 here you have the laundry equation and
59:55 then you have these terms here
59:57 on the right hand side there are higher
60:00 order
60:01 here we have this phi squared
60:04 that's just this one here and here
60:08 you have a noise term that we also had
60:10 before now that was this
60:12 gaussian noise term that we came from
60:14 integrating out the psi
60:16 in the martensitic rows formula now so
60:19 we have this term here
60:21 is quite noise but in contrast to the
60:23 previous case
60:25 where the noise didn't depend on the
60:27 density itself
60:28 we now here that's the only difference
60:31 have
60:31 another phi here
60:35 now we have another phi here
60:38 that's the only difference that we get
60:40 for multiplicative noise
60:42 and because we have multiplicative noise
60:46 you can see that somehow now the noise
60:50 term
60:51 here looks a little bit like another
60:55 term that is
60:55 actually as an interaction term where we
60:58 couple the two fields
61:00 one linearly to each other this term and
61:03 suddenly the noise term is not
61:04 simple a gaussian it's not simple a
61:07 gaussian that the inti can integrate
61:08 over
61:09 suddenly it couples to the other fields
61:11 now the strength of this noise curve
61:13 is proportional to phi that's this
61:16 multiplicative noise that makes life
61:18 complicated
61:20 now we now do a simple step now we
61:23 re-scale we now
61:24 we now we take this martensitic rows
61:27 generating function that i wrote down
61:29 here we take this
61:31 yeah and we just make our lives easier
61:33 life easier for later
61:35 yeah and by doing this to do this
61:39 we rescale some of these
61:42 fields and parameters so rescale
61:47 the fields so what we do
61:50 is we want to
61:53 get rid of
61:59 we have had two terms this one and this
62:02 one
62:03 they look kind of similar and the idea
62:06 is now
62:06 that if we have a proper transformation
62:09 or fields
62:10 that we can make them exactly equal up
62:13 to some pre-factors
62:15 yeah so that's what we want to do we
62:16 want to simplify this action
62:19 and uh to symmetrize it now so that we
62:22 can treat these two terms
62:23 and e equally now that's how we may we
62:26 want to make the prefactor here
62:28 this and this prefactor equal
62:32 now we can summarize these two trends
62:35 now so really scared the fields but just
62:37 tell you how to do that you have phi
62:40 goes over to 2 lambda
62:43 over gamma times phi
62:46 if i tilde the response field goes over
62:50 to
62:52 gamma to lambda
62:56 and gamma goes over to
63:00 two gamma lambda
63:05 yeah so we rescale this we are allowed
63:07 to do that
63:08 and then we get our new generating
63:11 functional
63:13 and this generating functional is again
63:16 of this form
63:17 d phi d phi tilde
63:22 e to the some action as naught and we
63:26 find that
63:26 what this is phi
63:30 by children
63:33 plus now this is has a naught the others
63:36 the non-interacting term
63:38 and now we have a term that of course
63:40 that describes interactions
63:43 now that's where the fun is happening
63:46 by artillery and just write that down
63:50 this as not phi
63:54 phi tilde this action is the integral
63:57 over dx
63:59 dt
64:03 so here we should have
64:06 dt as well
64:10 yeah so at the x dt
64:13 and now we um have
64:17 5 x t
64:21 tau sorry what we need that
64:25 del t minus d naught
64:28 squared minus kappa
64:32 phi of x t
64:37 and then we have another part as
64:40 interaction phi phi to the
64:45 gamma over two integral dx
64:49 dt phi tilde
64:53 x t minus i sorry that looks
64:56 not so nice
65:01 all right tilde of x and t
65:05 times phi of x
65:09 t minus phi to the
65:13 of x and t
65:17 phi of x
65:20 t so this is this interaction term
65:24 and it's called interaction term because
65:26 we have here
65:27 higher orders of the field coupling to
65:29 each other
65:31 yeah so
65:34 this is now our martin's intervals
65:37 integral
65:38 yeah and this is what we'll be dealing
65:41 with and this is what we'll define
65:43 the result renormalization group on and
65:46 because i knew that we would be doing
65:48 renormalization
65:50 i have already introduced this little
65:54 towel here now because in
65:57 renormalization
65:58 we want to cause grain we want to
66:00 transform this action
66:02 and we want to see how our action and
66:04 how the parameters are
66:06 of our action change in this procedure
66:09 and that's why all our terms here need
66:11 to have a prefactor
66:14 yeah and that's why i introduced this
66:15 tau that's why i have this d
66:18 you know and this kappa here i have them
66:20 all
66:21 giving them different names although
66:22 they're not independent of each other
66:25 because our original model just had one
66:27 parameter that was the lambda
66:29 yeah so that's the uh that's the martin
66:32 sutra rose
66:34 functional integral and um
66:38 next so we're already quite late now
66:41 next time
66:42 we start right away with uh first with
66:47 introducing renormalization intuitively
66:51 and then as a second step you'll then
66:53 apply that
66:54 to this epidemic model and re-normalize
66:57 this margin central rows
66:58 functional integral now apparently i was
67:01 always
67:01 very optimistic and it was trying to
67:04 introduce already
67:05 the renomination today but we can do
67:07 that just next week
67:08 now because it's already quite late
67:11 today