3. Non-equilibrium Field Theory

《非平衡态系统中的集体过程 (Collective processes in non-equilibrium systems)》是位于德累斯顿的马克思普朗克复杂物理研究所 (Max Planck Institute for the Physics of Complex Systems) Steffen Rulands 研究员的一门课程。

课程主页链接在此，网页上有课程的课件，录像发布于 YouTube。

YouTube 把视频中讲者说的话从语音转化成了文字，我把这些转录复制了下来，进行了简单的断句，并且推测了各段文字对应的课件的内容。

The slide 8 of examples was talked about at the end of the lecture rather than between neighboring slides.

random

`(00:00)` so i i have to set this sound to you so
`(00:02)` you hear them as well so you're using
`(00:04)` headphones okay right that goes
`(00:08)` okay so then let's start uh so welcome
`(00:11)` uh to our third lecture in collective
`(00:14)` processes and this will be
`(00:17)` the final lecture that is more
`(00:18)` methodology methodological
`(00:21)` and technical so today we'll talk about
`(00:24)` a field theory representations of
`(00:28)` the kind of processes that we've been
`(00:29)` looking at last time
`(00:32)` let me just share the screen
`(00:42)` okay

slide 1

`(00:49)` here we go so if you remember the
`(00:52)` lecture so you can see that
`(00:54)` great uh so you remember if you remember
`(00:56)` lecture that we had last time
`(00:59)` uh i gave a little introduction to the
`(01:02)` mathematics that is
`(01:04)` behind the description of the stochastic
`(01:07)` processes
`(01:09)` and we discussed two kinds
`(01:12)` of stochastic processes alternatives not
`(01:15)` two kinds of stochastic processes but
`(01:17)` two ways
`(01:18)` of describing sarcastic processes the
`(01:21)` first way
`(01:22)` that was associated with the name
`(01:24)` nonzero
`(01:25)` that we already featured in the very
`(01:26)` first lecture
`(01:28)` relied on deriving
`(01:32)` a stochastic differential equation that
`(01:34)` describes
`(01:36)` the time evolution of a single
`(01:38)` realization
`(01:39)` of such a stochastic process and the
`(01:41)` alternative approach that einstein
`(01:44)` applied for the description of broad
`(01:46)` emotion
`(01:47)` was to derive an equation
`(01:52)` for the time evolution of the
`(01:54)` probability density
`(01:56)` itself yeah and that was the master
`(01:58)` equation this master equation is not a
`(02:00)` stochastic
`(02:01)` differential equation is a deterministic
`(02:04)` equation
`(02:05)` but it's typically high dimensional and
`(02:09)` relies as you saw last time on some
`(02:13)` integrations over the possible states
`(02:16)` that you can jump into
`(02:18)` and then for those of you who remained
`(02:21)` for the example
`(02:23)` section last time you would have seen
`(02:26)` that these master equations although
`(02:29)` they look
`(02:30)` pretty complicated can be derived
`(02:32)` phenologically
`(02:34)` phenologically actually
`(02:37)` in most cases in a very simple way yeah
`(02:41)` so these are the two kinds of
`(02:42)` description and why are we not
`(02:44)` completely happy with that
`(02:48)` so there exists approximately
`(02:52)` suppose both in general conditions both
`(02:54)` of these different kinds of equations
`(02:55)` cannot be solved
`(02:57)` there exists approximative methods that
`(03:00)` we discussed last time for example the
`(03:01)` focal plunk equation
`(03:03)` that was an approximation for the master
`(03:06)` equation
`(03:07)` and these appropriate approximative
`(03:09)` methods that work in certain special
`(03:11)` cases
`(03:13)` now the focal planck equation as you
`(03:15)` remember the
`(03:16)` derivation relied on making strong
`(03:19)` assumptions
`(03:20)` on how these jumps in states state space
`(03:24)` look like they're very nicely behave
`(03:26)` these jumps are very small
`(03:28)` and that they effectively rely on very
`(03:32)` large
`(03:32)` system size and
`(03:36)` in other cases there are no
`(03:39)` approximative
`(03:40)` methods at all so what field theory
`(03:44)` does for us is it gives us a flexible
`(03:47)` framework a general
`(03:48)` and flexible framework that allows us to
`(03:51)` describe a large class of stochastic
`(03:54)` systems
`(03:55)` and it is even
`(03:59)` maybe if you remember from statistical
`(04:01)` physics or quantum mechanics
`(04:03)` it's uh so general that you can apply it
`(04:06)` to
`(04:06)` a very diverse set of systems that's
`(04:10)` one thing is for example spatially
`(04:12)` extended systems
`(04:13)` yes you can see on the bottom uh right
`(04:16)` yeah so this is a system where
`(04:20)` the evolution over the spatial
`(04:22)` information
`(04:23)` itself is very important not because you
`(04:26)` see here there's a chemical
`(04:28)` system you see that here structures form
`(04:31)` it's an example of what's called binodal
`(04:33)` decomposition
`(04:35)` now so here the spatial component is
`(04:36)` very important
`(04:38)` and this has to be taken into account
`(04:41)` when you want to understand of course
`(04:43)` collective processes
`(04:45)` another example are non-linearities in
`(04:48)` stochastic differential equations
`(04:51)` now these become especially if they
`(04:53)` affect an oyster and become pretty hard
`(04:55)` very quickly
`(04:56)` and another example is that i would have
`(04:58)` showed you last time
`(05:00)` that field theory that typical
`(05:02)` approximative methods like the focal
`(05:04)` planck
`(05:06)` method or other kinds of expansions
`(05:10)` uh are not very suitable for rare events
`(05:14)` for the tales of probability
`(05:16)` distributions
`(05:17)` so in fields here we have a general
`(05:19)` framework
`(05:20)` of understanding these variables
`(05:24)` and so there has been a lot of work on
`(05:26)` how to make approximations to these
`(05:28)` field theories
`(05:30)` that allow us to understand the tales of
`(05:32)` probability distributions and very often
`(05:34)` these tales
`(05:36)` are pretty important so they are not
`(05:39)` they're rare
`(05:40)` but uh if you think about a nuclear
`(05:44)` reactor or something like this yes these
`(05:46)` rare events are rare
`(05:47)` but you want to know how often they hear
`(05:49)` and there are a few theoretic methods
`(05:51)` that allow you
`(05:52)` to calculate the tails of probability
`(05:54)` distributions
`(05:56)` very nicely so
`(06:00)` this is what we'll do and in this
`(06:02)` lecture

slide 2

`(06:03)` i want to show you how we can derive
`(06:07)` a field theory description um
`(06:11)` for the longitudinal equation for
`(06:13)` stochastic differential equations
`(06:15)` we'll be looking at very simple a
`(06:20)` very simple automatic equation that
`(06:22)` doesn't have any
`(06:23)` explicit time derivative that doesn't
`(06:25)` have
`(06:26)` multiplicative noise and we can
`(06:30)` nevertheless in the framework of field
`(06:31)` theory we can
`(06:33)` straightforwardly uh extend
`(06:36)` this methodology to more complicated
`(06:38)` systems so we restrict ourselves
`(06:40)` to this very simple larger equation uh
`(06:43)` with gaussian white noise
`(06:45)` that is as last time uncorrelated
`(06:50)` so if you
`(06:54)` so the first step here is to discretize
`(06:58)` time yeah and uh so we take
`(07:02)` the logical equation and we write it
`(07:03)` down in discrete time
`(07:06)` and uh as you saw uh last time
`(07:09)` uh or in the first of what was the first
`(07:12)` lecture already i saw
`(07:13)` two lectures ago oh wait whatever yeah
`(07:16)` so so no last time last time was the
`(07:18)` catholic processes
`(07:19)` yeah so as you saw last time the way we
`(07:22)` discretize
`(07:24)` stochastic differential equations and i
`(07:27)` showed you that for uh stochastic
`(07:28)` integrals is very important
`(07:32)` yeah and in this case we also have to
`(07:34)` discretize
`(07:35)` our stochastic differential equation in
`(07:37)` a certain way
`(07:39)` which is called the ito discretization
`(07:43)` and but if we do that if we discretize
`(07:47)` our stochastic differential equation in
`(07:49)` this way
`(07:50)` the idea is that in principle we can
`(07:53)` write down
`(07:55)` averages over any
`(07:58)` observable you know that's here on the
`(08:01)` left left hand side by some complicated
`(08:04)` thing that is on the right hand side
`(08:07)` so this what is on the right hand side
`(08:09)` and this equation looks pretty
`(08:10)` complicated but it's not that
`(08:12)` complicated so let's have a look
`(08:14)` at these different terms in the first
`(08:18)` term here
`(08:19)` on the left right hand side the red term
`(08:22)` we just integrate over all possible
`(08:25)` realizations
`(08:27)` that a stochastic trajectory can take
`(08:32)` on the right hand side
`(08:35)` now if you look at the right hand side
`(08:37)` oh there's a delta function missing here
`(08:40)` right at here delta here
`(08:43)` on the right hand side you have this
`(08:45)` delta function
`(08:49)` here you have this delta function and
`(08:52)` this delta function
`(08:54)` just makes sure that whatever we
`(08:56)` integrate here whatever trajectories we
`(08:59)` integrate over
`(09:01)` that they fulfill the discretized
`(09:04)` version
`(09:05)` of the larger equation
`(09:08)` yeah so the delta function is just the
`(09:11)` left-hand side
`(09:12)` minus the right-hand side of this launch
`(09:15)` of our equation
`(09:16)` and if the left-hand side is equal to
`(09:18)` the right-hand side
`(09:20)` then we take that trajectory into
`(09:22)` account
`(09:23)` you know and then sandwiched between
`(09:26)` that we have
`(09:27)` our observable o that is some functional
`(09:31)` of our trajectory x
`(09:34)` now as i say functional so i don't know
`(09:37)` how much uh
`(09:37)` functional analysis all of you had so
`(09:40)` most
`(09:41)` so i would expect that most of you would
`(09:44)` have that
`(09:44)` until the first fourth term
`(09:48)` or so also but just just to make sure a
`(09:51)` functional
`(09:52)` is basically a function that maps um
`(09:56)` that maps this that maps
`(09:59)` this function x uh yes that map
`(10:03)` a function to a real number yeah
`(10:06)` and this is our how we define these
`(10:08)` observables
`(10:09)` now you take a trajectory and you map it
`(10:12)` to them to a number
`(10:16)` okay so this looks very complicated it
`(10:18)` doesn't help us
`(10:19)` anything at all and uh
`(10:22)` one the other step that we need to make
`(10:24)` on the slide is we that we
`(10:26)` introduce some notations here
`(10:29)` and uh of course we don't always want to
`(10:32)` write
`(10:33)` all of these integrals here on the left
`(10:36)` hand side
`(10:37)` we just say we write that in this
`(10:39)` functional form here
`(10:41)` and of course many of you will know that
`(10:43)` this is just
`(10:44)` a functional integral or a path into
`(10:47)` yeah that's how we define it here

slide 3

`(10:49)` and now for uh notational convenience
`(10:54)` now we just go to a continuum locate
`(10:56)` notation now we forget that we
`(10:58)` discretize time in the previous step
`(11:01)` and we write the equation back in
`(11:03)` continuous form
`(11:05)` just for the sake of simplicity and we
`(11:07)` define
`(11:08)` some delta functional
`(11:11)` that is just equal to the product over
`(11:14)` all data functions that we had on the
`(11:16)` previous slide
`(11:17)` now if you look here are the product of
`(11:20)` our data functions
`(11:21)` just make sure that you fulfill the
`(11:23)` larger equation really at each time
`(11:25)` point
`(11:27)` and we just define like a super delta
`(11:29)` function or delta functional
`(11:31)` that makes sure that we really satisfy
`(11:33)` the larger equation for each time point
`(11:41)` so now we make a little trick
`(11:45)` now we say we have this delta function
`(11:47)` or this product of delta functions
`(11:50)` and what we say is that in fourier space
`(11:54)` the delta function is represented by a
`(11:58)` plane
`(11:58)` wave yeah so we fully transform the
`(12:02)` delta function
`(12:15)` or a functional is now and
`(12:18)` the fourier transform of the delta
`(12:20)` function
`(12:21)` just becomes delta x dot
`(12:25)` minus f of x minus
`(12:29)` sine because our delta our function
`(12:31)` equation
`(12:33)` is equal to a fury space
`(12:38)` d x tilde
`(12:43)` e to the minus i x to the
`(12:48)` x dot minus
`(12:52)` f of x minus c
`(12:56)` now we now get this variable x tilde
`(13:00)` so this is nothing there's nothing
`(13:01)` happening it's just the definition of
`(13:03)` the forage in the form
`(13:04)` of a delta function and because we have
`(13:08)` this
`(13:08)` product for many delta functions here
`(13:12)` we have the integral here that the path
`(13:15)` integral here will also get a path
`(13:17)` integral
`(13:18)` delta x total you know that's that just
`(13:22)` the definition of the fury transform
`(13:24)` and now we plug this
`(13:28)` in again so
`(13:32)` we obtain from that
`(13:35)` that the expectation value of our
`(13:39)` observable yeah
`(13:43)` taken to the average now this average is
`(13:45)` over the distance
`(13:47)` over over a different voice realizations
`(13:52)` yeah it is equal now we plug that in
`(13:55)` an integral now over
`(13:58)` x and x tilde so it's a path integral
`(14:03)` over x and x tilde so we there's an
`(14:06)` integral
`(14:07)` over all realization of x and all
`(14:10)` realization
`(14:11)` of x2 now so now this x still that pops
`(14:14)` up yeah so we don't really know what it
`(14:15)` is
`(14:16)` but it will hang around and it will show
`(14:18)` you later what it actually mean
`(14:21)` means so we have this fourth integral
`(14:25)` so our observable of x and then
`(14:33)` plugging in e to the minus
`(14:37)` i x tilde
`(14:40)` um sorry
`(14:44)` we've got an integral so we have here a
`(14:46)` little
`(14:47)` integral dt
`(14:52)` and this integral
`(14:56)` we get because we had this
`(15:00)` product over here now so here the
`(15:03)` product
`(15:04)` we had here gives us an integral so our
`(15:07)` sum
`(15:08)` in the exponential so this was
`(15:10)` originally like a product of many
`(15:12)` exponential
`(15:14)` and uh so this is this and then we have
`(15:18)` our
`(15:18)` x tilde x dot minus
`(15:23)` f of x minus x psi
`(15:28)` and then we close the average
`(15:32)` and then we just plug this in yeah we
`(15:34)` can move out
`(15:36)` everything that does not depend on psi
`(15:39)` or the noise
`(15:41)` out of the average because average is
`(15:43)` over the noise
`(15:46)` b x x to the
`(15:51)` power of x right now comes the stuff
`(15:54)` that does not depend
`(15:56)` on x under x i integral
`(16:00)` e t x to the
`(16:04)` x dot minus f of x
`(16:08)` and now we have some
`(16:12)` average over minus i
`(16:15)` dt x tilde
`(16:19)` times psi

slide 4

`(16:23)` you know so
`(16:26)` now the question is what is this here
`(16:32)` can we calculate this at the moment we
`(16:34)` cannot do anything with this equation
`(16:36)` with this expression can you can we
`(16:38)` calculate
`(16:40)` this last term that involves a noise
`(16:44)` average
`(16:45)` over e to the minus dt
`(16:49)` and psi now and there's hope that we can
`(16:52)` calculate this because we know what
`(16:54)` psi is now we said that x i
`(16:58)` is a gaussian random variable
`(17:01)` yeah it has follows a normal
`(17:03)` distribution
`(17:04)` it's uncorrelated so we know a lot of
`(17:07)` things about this
`(17:08)` sign and what we do right now
`(17:12)` right because psi is gaussian there's
`(17:14)` also hope that we can actually
`(17:17)` solve or integrate this integral that is
`(17:20)` this average here
`(17:21)` now i'll show you now how this works in
`(17:24)` detail
`(17:25)` yeah so we make use of the definition of
`(17:28)` phi
`(17:30)` so let's see what this second average
`(17:32)` looks like
`(17:34)` now that's o of x
`(17:38)` uh sorry it's not o of x it is
`(17:42)` e to the minus i dt
`(17:47)` integral x of
`(17:50)` sine
`(17:54)` now and this is
`(17:58)` by definition by the definition of the
`(18:00)` average
`(18:03)` is equal to the integral over dxi
`(18:08)` times the probability over the
`(18:11)` probability distribution of
`(18:13)` psi which is a gaussian or a normal
`(18:15)` distribution
`(18:18)` 2 pi a a is the strength of our noise
`(18:23)` e to the minus psi
`(18:26)` squared over 2a
`(18:29)` yeah so this is the probability
`(18:31)` distribution
`(18:33)` and we know that psi is normally
`(18:36)` distributed
`(18:37)` yeah and that's why we have this normal
`(18:40)` distribution here
`(18:42)` at now we multiply this by the thing
`(18:45)` we're averaging over
`(18:48)` yeah e to the minus i
`(18:51)` d t x to the sine
`(18:57)` now so this excuse me yes isn't it
`(19:00)` supposed to be exponential
`(19:02)` plus psi integral delta t whatever
`(19:08)` integral over let me see
`(19:12)` you mean the second integral this one
`(19:13)` here
`(19:16)` yeah okay so this one is that minus or
`(19:18)` plus
`(19:20)` let me just check
`(19:23)` let me check my notes
`(19:27)` okay
`(19:31)` so here we go
`(19:35)` so no there's a minus
`(19:39)` i i wouldn't be able to find him i
`(19:42)` wouldn't be able to find it
`(19:43)` [Music]
`(19:45)` on the previous slide you can go and
`(19:47)` just
`(19:48)` okay so let me see maybe there's a here
`(19:51)` we have the minus
`(19:54)` here we have the minus u of the minus
`(19:57)` here we have the minus and another minus
`(20:00)` yeah you're right
`(20:01)` let me see what's wrong here uh
`(20:04)` okay oh yes okay so here
`(20:12)` i think this minus here is not right
`(20:17)` now that's just the definition here wait
`(20:20)` minus my i don't know
`(20:22)` okay minus minus is plus
`(20:28)` minus minus with plus i think i think
`(20:29)` you're right
`(20:31)` but then here should be a minus so where
`(20:33)` does the
`(20:34)` is there an i squared somewhere
`(20:44)` excuse me i think there should be a
`(20:46)` minus sign

slide 4

`(20:47)` um in the first exponential as well in
`(20:50)` the last step on this particular slide
`(20:52)` there are two exponential studies yes
`(20:54)` yes yes yes yes
`(20:56)` that's what i'm wondering about um let
`(20:58)` me see if it's
`(20:59)` actually the final result
`(21:02)` um i x minus minus
`(21:08)` [Music]
`(21:14)` okay so so mike
`(21:17)` let me let me just see
`(21:24)` let me just see
`(21:28)` so let's let's put this minus here in
`(21:30)` brackets
`(21:31)` now so this one minus this should have
`(21:34)` the opposite
`(21:35)` i don't wait with me okay this is the
`(21:37)` minus then
`(21:38)` this should be plus you say
`(21:42)` you know i think i think it i think it
`(21:45)` will come out correctly later
`(21:46)` now let's see how it goes um let's see
`(21:50)` how it goes
`(21:51)` now it makes sense
`(21:56)` now we have to have a gaussian integral
`(22:00)` and that will determine whether what
`(22:04)` we're doing is right
`(22:05)` okay so let's let's go on so this here
`(22:08)` is the integral and now we just
`(22:12)` put everything together and say
`(22:15)` that the sign 1 over
`(22:20)` 2 a e
`(22:24)` minus dt x to the
`(22:28)` psi
`(22:32)` now that's the first one so that would
`(22:34)` be a plus then
`(22:36)` minus i dt x
`(22:40)` to the sine
`(22:43)` now i just i just uh i just combined the
`(22:46)` exponentials
`(22:47)` now thanks for thanks for paying so much
`(22:49)` attention
`(22:51)` um so yeah and this is here
`(22:54)` a gaussian integral and if you don't
`(22:57)` know remember
`(22:58)` the gaussian integrals and you can look
`(23:00)` at the bottom here
`(23:01)` these gaussian intervals are pretty easy
`(23:04)` to solve
`(23:05)` probably most of you have heard of that
`(23:08)` and we can also solve
`(23:10)` this gaussian integral here and what we
`(23:13)` get
`(23:13)` is that this is just e to the a over 2
`(23:18)` dt x to the squared
`(23:23)` yeah because we know yeah and here
`(23:27)` one thing you have to make uh you need
`(23:29)` to remember
`(23:30)` is that this here has eyes in it
`(23:33)` now if you look at this formula at the
`(23:35)` bottom you need to take into account
`(23:36)` that these are complex
`(23:38)` integrals so we can calculate
`(23:42)` this quantity here
`(23:45)` we can calculate this quant this
`(23:46)` quantity here because we know
`(23:49)` how psi looks like and that we integrate
`(23:52)` over the distribution of
`(23:53)` sine of the noise organization and we
`(23:56)` get
`(23:56)` the term that we have here yeah
`(23:59)` and now we already have arrived at the
`(24:02)` famous
`(24:03)` no at least famous for a very small
`(24:05)` number of people

slide 5

`(24:06)` and this is the so-called martin citra
`(24:09)` rose johnson they dominicus
`(24:12)` uh functional integral yeah that's what
`(24:14)` you see here
`(24:16)` in the red box now we just now put
`(24:18)` everything together
`(24:19)` here we have our noise here we have
`(24:23)` uh what we had before and here
`(24:28)` is so to say that it's a deterministic
`(24:30)` part that came from the larger equation
`(24:33)` yeah and uh so what this tells us now
`(24:37)` here
`(24:38)` and maybe it looks familiar to some of
`(24:40)` you yet quantum field theory
`(24:42)` this looks very familiar so what we do
`(24:45)` now is
`(24:46)` we want to calculate the average over
`(24:49)` some observable
`(24:52)` we integrate over all possible
`(24:56)` realizations and over all possible
`(24:58)` realizations of some weird quantity
`(25:00)` x tilde so we got a second field here
`(25:05)` and weight the contributions of
`(25:08)` different trajectories
`(25:11)` by this exponential factor here
`(25:14)` now and this exponential factor looks
`(25:16)` very much like what you know from
`(25:19)` other field theories like quantum field
`(25:21)` theory and this is why this is very
`(25:23)` often called
`(25:24)` an action
`(25:28)` now that's the martin sutra rose or ms
`(25:31)` rjd functional
`(25:34)` integral that allows us to really write
`(25:37)` a field theory
`(25:39)` for stochastic processes
`(25:42)` now so here our axes are not fields yet
`(25:45)` now they're trajectories they don't have
`(25:47)` a space component now they don't have
`(25:49)` spatial dimensions
`(25:51)` but as you can see later the structure
`(25:53)` of a real spatial
`(25:55)` the the integrals they will look very
`(25:58)` similar
`(26:00)` now we can also rewrite this a little
`(26:04)` bit here and look at specific
`(26:05)` trajectories and then we can look for
`(26:09)` example at a special case
`(26:11)` where it is observable is just
`(26:14)` the propagator here or this will
`(26:16)` propagate as the probability
`(26:18)` that we end up at some state x at a time
`(26:22)` t
`(26:23)` if we start it add some x naught
`(26:27)` at a time t naught you know and we
`(26:30)` obtain this
`(26:31)` not by just requiring that this
`(26:33)` observable is a delta function
`(26:36)` where uh we say that the x the specific
`(26:39)` time
`(26:40)` t at a time t needs to be equal
`(26:43)` to the x that we give to the probability
`(26:46)` distribution here
`(26:50)` and then we plug that in and
`(26:53)` what we now need to say is that we only
`(26:56)` integrate
`(26:57)` over trajectories that actually started
`(27:00)` it's not an end
`(27:01)` and x at a given times now that's that's
`(27:04)` what we have to take to account for in
`(27:06)` the boundaries and the bounds of the
`(27:07)` integrals
`(27:08)` yeah and then we get this form here
`(27:13)` now that looks very similar and that's a
`(27:16)` different representation if you remember
`(27:18)` last time we had the
`(27:19)` koi maguro of uh chap and komogorov
`(27:21)` equation
`(27:22)` it was an equation for the same quantity
`(27:25)` and the idea was a little bit similar
`(27:27)` in this technical mcgovern
`(27:30)` equation we had the same spirit
`(27:33)` and we looked at different sums of
`(27:35)` different paths
`(27:37)` um a process could take to go from x
`(27:40)` naught to x
`(27:41)` and here we do the same thing in a more
`(27:43)` fancy way
`(27:49)` just to reflect on this a little bit
`(27:51)` more
`(27:52)` so what happened now so we started
`(27:55)` here with
`(27:59)` a longer equation we discretized it
`(28:03)` and if you remember uh quantum fields
`(28:06)` theory that's also the step that you do
`(28:08)` there
`(28:08)` you discretize time into small intervals
`(28:12)` as the first step if you derive
`(28:13)` a quantum fluid theory and then
`(28:17)` we wrote expectation values of some
`(28:19)` observables
`(28:20)` formally in a way that involved
`(28:23)` integrals of
`(28:23)` all possible trajectories and the
`(28:25)` resultant trajectories
`(28:27)` filtered four trajectories that solve
`(28:30)` the launch of a creation
`(28:32)` now that was to say self uh
`(28:35)` circular starting point and in the next
`(28:39)` uh step we then got this field
`(28:42)` side tilde now that we still don't know
`(28:44)` what it is exactly about
`(28:46)` now we got that from the fourier
`(28:47)` transform of the delta function
`(28:52)` and now now we have two fields we
`(28:54)` integrate over two fields
`(28:56)` x and x x tilde and
`(28:59)` in the end we managed to integrate out
`(29:03)` the noise so what we have here
`(29:07)` now is something that does not depend on
`(29:09)` the sign anymore
`(29:11)` it's a deterministic
`(29:14)` equation and deterministic integral so
`(29:17)` somehow
`(29:18)` our noise was observed observed absorbed
`(29:23)` into a new field a new fluctuating field
`(29:28)` x tilde yeah and that's how
`(29:31)` it very often goes now that you
`(29:34)` make a field theory and what you gain is
`(29:37)` you get a nice nice integral but you
`(29:39)` have to pay for
`(29:40)` it by having additional fields conjugate
`(29:43)` fields
`(29:44)` that you have to integrate over and the
`(29:47)` same is true here
`(29:48)` and i'll show you later what these x
`(29:50)` tilde actually mean before i do that let

slide 6

`(29:55)` me just mention so so now we have a few
`(29:56)` theory i have a path integral and these
`(29:59)` path integrals are very useful
`(30:01)` because we can make use of a lot of
`(30:03)` tools
`(30:04)` from other field fields from quantum
`(30:06)` field theory
`(30:08)` renormalization perturbation theory and
`(30:10)` so we can
`(30:11)` make use of these tools very powerful
`(30:14)` frameworks developed in the last 70
`(30:18)` years or so
`(30:20)` we can make use of these frameworks and
`(30:21)` apply them to these
`(30:23)` few theories for stochastic processes
`(30:27)` and one of the things that we can do is
`(30:29)` we can
`(30:30)` define a so-called generating functional
`(30:35)` that's maybe something that you already
`(30:36)` know from other field theories
`(30:38)` so what you do is you add some auxiliary
`(30:42)` fields
`(30:43)` external fields h and h
`(30:46)` tilde and these fields cuddle
`(30:51)` to x and x tilde the accelerations
`(30:55)` respectively
`(30:57)` and what is this is then the generating
`(31:00)` function
`(31:01)` and of course we know that these fields
`(31:04)` don't really exist
`(31:05)` now we just added them and the reason
`(31:08)` why we added them
`(31:10)` is that if we take derivatives
`(31:13)` with respect to these fields h or this
`(31:16)` external
`(31:17)` forces or external fields h here
`(31:21)` and here what will happen is that each
`(31:24)` time
`(31:25)` because this is an in exponential
`(31:28)` the x
`(31:31)` let or just draw that now the x
`(31:36)` or the x tilde
`(31:40)` will go down here
`(31:44)` yeah and if we do that if we take these
`(31:46)` derivatives
`(31:48)` you know so we can therefore get the
`(31:52)` expectation values of combinations of
`(31:56)` the x
`(31:56)` and x tildes just by differentiating
`(31:59)` this
`(32:01)` generating functional with respect to
`(32:04)` these
`(32:05)` weird virtual fields
`(32:08)` yeah and when we've done that we have to
`(32:12)` remove these fields again so we have to
`(32:14)` set them back to zero
`(32:15)` now for example if you want to have the
`(32:17)` correlation the autocorrelation function
`(32:19)` so how much
`(32:21)` is the process at a time t correlated to
`(32:24)` the state at a time t
`(32:25)` prime yeah then we
`(32:28)` differentiate and we
`(32:32)` take the derivative first with respect
`(32:34)` to
`(32:35)` with respect to h at a certain time t
`(32:39)` yeah and then we get one of these axes
`(32:42)` here and then we take the derivative
`(32:44)` with uh to h at
`(32:48)` a time a different time t prime and then
`(32:50)` we get the field again
`(32:52)` at a different time here
`(32:55)` yeah and these are of course functional
`(32:57)` derivatives now that was
`(32:59)` uh functional calculus if you haven't
`(33:01)` done that
`(33:02)` it's uh for these what you what you do
`(33:04)` is you
`(33:05)` look at the change of a functional
`(33:08)` now for example this here is a
`(33:10)` functional
`(33:12)` we look at the change of a functional
`(33:15)` with respect to small
`(33:16)` changes of its argument also you have a
`(33:20)` look at
`(33:20)` perturbations in age and h tilde
`(33:24)` around some value yeah and then you
`(33:28)` see how your function changes that's
`(33:30)` called a functional derivative
`(33:32)` and if you do that you get very
`(33:34)` conveniently these pre-factors here
`(33:39)` right here in front of the action and if
`(33:42)` you look at
`(33:43)` the definition here this is just what
`(33:46)` gives us our observable
`(33:48)` as for example if our observable is x
`(33:52)` that we just take the derivative once
`(33:55)` we get an x here yeah
`(33:58)` and if we have the x here we have the
`(34:01)` first moment so the mean
`(34:03)` of x the average of x if we take the
`(34:06)` derivative twice
`(34:07)` at different times then we get a
`(34:09)` correlation here on the left-hand side
`(34:13)` so this is a very convenient tool and as
`(34:15)` i said
`(34:16)` if you want to have a correlation
`(34:18)` function for example we take the
`(34:19)` derivative
`(34:20)` twice and you must always remember to
`(34:23)` set
`(34:23)` these fields to zero again it's actually
`(34:26)` the same approach as you do in the
`(34:28)` quantity
`(34:29)` and classical field theory the
`(34:31)` equilibrium equilibrium field t
`(34:37)` okay so now what is this x
`(34:40)` of t that's just a remark so i'm not
`(34:42)` doing the calculations like

slide 7

`(34:44)` what is this x tilde of t that we get
`(34:48)` got in this process
`(34:52)` there are different uh
`(34:55)` c theories for stochastic processes and
`(34:57)` for master equations
`(34:58)` and you always get some kind of
`(35:01)` auxiliary
`(35:02)` field some some conjugate field that you
`(35:05)` have to pay for
`(35:07)` and uh this x tilde from here from here
`(35:10)` you can
`(35:12)` get an intuition about that i'm not
`(35:14)` super rigorous but you can get
`(35:16)` your intuition about this if you write
`(35:18)` down the master equation
`(35:20)` sorry the larger equation with respect
`(35:24)` and add some external source
`(35:27)` capital h of t
`(35:32)` you know and if you add this external
`(35:34)` force capital h
`(35:36)` of t now some temporally fluctuating
`(35:39)` force
`(35:40)` that doesn't depend on x itself
`(35:43)` and you plug that in into this martin
`(35:46)` sergio rose
`(35:47)` functional integral or martin citra ruse
`(35:50)` johnson did you limit it to minikit
`(35:54)` then you see a formal analogy that if
`(35:57)` you
`(35:57)` that this you get a term that looks like
`(36:00)` h tilde
`(36:04)` if you define this to be minus i this
`(36:07)` external field
`(36:09)` yeah and now we can see what happens
`(36:14)` to x what is the effect of this external
`(36:17)` field
`(36:19)` h of t on x so there's a little bit in
`(36:22)` an
`(36:23)` analogy already here now so here on the
`(36:25)` left hand side it has something to do
`(36:27)` with this uh external field from the
`(36:31)` generating functional that couples to x
`(36:33)` tilde
`(36:34)` now let's see what this does to x to the
`(36:37)` actual stochastic process
`(36:39)` now to this end we calculate a response
`(36:43)` function
`(36:43)` and this response function is just the
`(36:47)` change
`(36:48)` in the average of f x with respect
`(36:52)` to changes in this external field
`(36:55)` that's called the response so how does
`(36:56)` the system respond
`(36:58)` to changes in this external field
`(37:01)` yeah and so as you remember the average
`(37:05)` here we just get by
`(37:08)` integrating uh by by taking this for
`(37:12)` this this generating functional
`(37:15)` and taking the derivative with respect
`(37:17)` to h
`(37:18)` once so that's the first moment
`(37:22)` and because of this equality here
`(37:27)` now we see that this year
`(37:30)` this response function is we also get
`(37:33)` that
`(37:33)` if we take the derivative with respect
`(37:36)` to h
`(37:37)` and then h tilde at a certain time
`(37:41)` t let me see let me just say
`(37:45)` here that's the tilde
`(37:51)` yeah and this here what this is
`(37:59)` as on the last slide is the correlation
`(38:02)` between
`(38:02)` x of t and x tilde of t
`(38:06)` now somehow the response of the system
`(38:10)` with respect to an external force
`(38:13)` or an infinitely miserable external
`(38:16)` force
`(38:17)` is given by how the x total
`(38:21)` couples to x
`(38:24)` so it describes the activity or is
`(38:27)` related
`(38:28)` to an infinitesimal response
`(38:32)` of x of the field x with respect
`(38:35)` to a small perturbation and that's why
`(38:39)` this field x tilde is also called the
`(38:41)` response field
`(38:44)` now there's just a little bit of
`(38:46)` intuition and very often this these
`(38:48)` conjugate variables somehow in some way
`(38:51)` encode the noise
`(38:56)` now i have an example uh let me just
`(38:59)` check the time whether we do it now or
`(39:01)` at the end of the lecture
`(39:05)` let's do it at the end of the lecture
`(39:06)` again not this example for those of you
`(39:09)` or who
`(39:10)` heard a few theory course last year uh
`(39:13)` there were no
`(39:14)` examples so i'll leave that to the to
`(39:16)` the end of the lecture that
`(39:17)` people can dial out uh if they're tired

slide 9

`(39:24)` now i want to just give you a second
`(39:26)` remark
`(39:28)` so we are now in the framework of
`(39:29)` physicians
`(39:31)` and in field theory now if you remember
`(39:34)` we can have different formulations of
`(39:36)` field theories otherwise the lagrangian
`(39:39)` field theory and the hamiltonian field
`(39:42)` and we can translate these two into each
`(39:45)` other
`(39:45)` and we can do the same things here for
`(39:49)` the non-equilibrium for the stochastic
`(39:51)` process
`(39:53)` yeah and it's also just a remark no it's
`(39:56)` not
`(39:57)` so important for the rest of the lecture
`(40:00)` but you can formally define
`(40:03)` new variables q and
`(40:07)` p by these relations here
`(40:10)` and then this probability will take the
`(40:14)` form
`(40:14)` that i wrote down here now so this
`(40:18)` has formally the form of a hamiltonian
`(40:22)` action oh or having a hamiltonian theory
`(40:25)` so we have
`(40:26)` p times q dot minus
`(40:29)` some hamiltonian and this hamiltonian
`(40:32)` is given by this term here this looks a
`(40:35)` little bit like a kinetic energy so it's
`(40:38)` just
`(40:38)` just just saying that we can write by
`(40:41)` variable transformation we can write
`(40:42)` these
`(40:43)` field theories uh then quite analogously
`(40:47)` to feed theories that we already know
`(40:51)` and we can even go one step further
`(40:55)` now we can take this here and integrate
`(40:58)` out the piece
`(40:59)` now because it's just gradually
`(41:01)` quadratic in p
`(41:03)` so these are just essentially gaussian
`(41:05)` integrals that you can integrate over
`(41:06)` them
`(41:07)` and just to tell you the results that we
`(41:09)` then get
`(41:10)` a field theory that looks like a
`(41:13)` lagrangian field theory
`(41:16)` now we have a lagrangian that depends on
`(41:17)` q and the
`(41:19)` derivative of q with respect to time
`(41:22)` and if you then look at the analogy
`(41:26)` at this hammer at this land range here
`(41:28)` then
`(41:29)` this looks like it describes some kind
`(41:31)` of particle
`(41:32)` with that has some mass one over a uh
`(41:35)` that lays
`(41:36)` like in a potential that happens that's
`(41:38)` coupled to some
`(41:39)` q dot here and uh we have now here
`(41:43)` the quadratic potential of laughter now
`(41:46)` so now these analogies they're not very
`(41:48)` helpful yeah they don't tell you
`(41:49)` anything
`(41:50)` uh these hamiltonians that you get here
`(41:53)` uh
`(41:53)` they're not comparable to hamiltonians
`(41:55)` that you get important systems
`(41:57)` for example these hamiltonians they're
`(42:01)` not emission quantities
`(42:05)` for quantum physicists
`(42:10)` and just to say that they're different
`(42:11)` ways of formulating these
`(42:13)` theories that you get by variable
`(42:16)` transformations
`(42:17)` uh this second equation here has a name
`(42:19)` that's the own sagar
`(42:21)` mac look functional you know and like
`(42:24)` you will
`(42:25)` pop up these these these kind of
`(42:27)` functional
`(42:28)` um pop up in papers if you read papers
`(42:31)` that
`(42:32)` they they pop up in different ways but
`(42:34)` in the end
`(42:35)` the same uh um
`(42:40)` implementations as a few implementations
`(42:43)` of the same
`(42:44)` stochastic differential equation but
`(42:46)` just reformulations of the same thing

slide 10

`(42:52)` okay now
`(42:55)` i told you in the beginning of the
`(42:57)` lecture that um
`(43:01)` i actually what most of the cases
`(43:03)` interested in
`(43:04)` now and what field theories are also
`(43:07)` good for
`(43:08)` are spatially extended systems
`(43:13)` now how do you write down a larger
`(43:16)` equation or a spatially extended system
`(43:19)` now it gets of course a little bit more
`(43:20)` complicated but there's actually a
`(43:22)` classification
`(43:23)` theme for a long general equation that
`(43:27)` describes systems
`(43:28)` that have spatial degrees of freedom
`(43:32)` and this classification scheme you can
`(43:34)` see on the slide
`(43:36)` and so we're looking here at some time
`(43:40)` evolution of some field phi
`(43:43)` at a position x at a time t
`(43:47)` and this time evolution is given by
`(43:49)` different components
`(43:51)` yeah that's r on the right hand sides on
`(43:54)` the very right we have the noise
`(43:56)` as always yeah and this noise
`(44:01)` now has the usual properties the average
`(44:04)` of the noise is zero
`(44:06)` and it also has correlations so how is
`(44:09)` a fluctuation a fluctuating force of
`(44:12)` that xi represents how is that
`(44:15)` correlated between different time points
`(44:18)` and how they correlated between
`(44:20)` different positions
`(44:21)` in space so while the assumption that
`(44:24)` different
`(44:25)` correlations that this noise is
`(44:28)` uncorrelated in time so that
`(44:30)` you don't have memory is something that
`(44:32)` is
`(44:33)` very reasonable and that we that we
`(44:35)` typically make
`(44:37)` it's not so clear that the noise is also
`(44:40)` independent at each position in space
`(44:44)` you know so generally there will be some
`(44:47)` some
`(44:48)` length you perturb the system and then
`(44:49)` you don't perturb a single atom
`(44:51)` yeah but you deter like a small region
`(44:54)` for example
`(44:55)` and then these fluctuating forces are
`(44:58)` correlated
`(45:00)` over small regions and these
`(45:02)` correlations in the spatial
`(45:06)` uh in these spatial systems and the
`(45:09)` fluctuating force that act on these
`(45:11)` systems
`(45:13)` are given by so-called spatial
`(45:15)` correlation
`(45:16)` now that just says how are these
`(45:19)` fluctuating forces
`(45:21)` how are they correlated between two
`(45:24)` given positions
`(45:25)` in space yeah and
`(45:28)` now we have the rest here
`(45:32)` the black parts here
`(45:35)` yeah it's just a fancy way of writing
`(45:38)` down
`(45:39)` the deterministic part of what's
`(45:42)` actually happening
`(45:44)` yeah and it's just a fancy way of
`(45:47)` writing down uh the actual
`(45:50)` uh for example chemical reactions
`(45:53)` that change the value of the field at a
`(45:56)` given position
`(45:58)` yeah and we can formally write down this
`(46:01)` uh this term by saying okay so we have
`(46:04)` some kind of potential
`(46:06)` and the dynamics will go so to some
`(46:09)` minimum
`(46:10)` of this potential not just some some
`(46:13)` equilibrium state
`(46:15)` yeah and then uh we can formally write
`(46:18)` this
`(46:18)` this way here that we have some
`(46:21)` functions and free energy functional
`(46:23)` f that depends on the fields and this is
`(46:26)` given by some input
`(46:28)` space integral uh where we have this
`(46:31)` term here that will flatten
`(46:33)` the field that something like can become
`(46:36)` a diffusion term
`(46:37)` and on the right hand side we have
`(46:40)` something
`(46:41)` that is a potential that describes
`(46:43)` what's actually which where we're going
`(46:44)` to with the field
`(46:46)` now if you have to have heard
`(46:48)` statistical physics
`(46:50)` yeah then uh these will look familiar to
`(46:53)` them
`(46:53)` to you like fight to the force theory
`(46:55)` ginsburg londo and so on
`(46:58)` now there's this red stuff here yeah and
`(47:01)` this red stuff is just a way
`(47:03)` of classifying different kinds of
`(47:06)` systems
`(47:07)` and people distinguish between system
`(47:10)` where the order parameter so our phi
`(47:13)` is not conserved for example in chemical
`(47:16)` systems where you can
`(47:17)` convert one chemical species to another
`(47:20)` chemical species
`(47:21)` yeah and then another chemical species
`(47:23)` and so on then the con
`(47:25)` concentration of one chemical species is
`(47:28)` not
`(47:28)` conserved in the entire system
`(47:32)` yeah and this is what you get if you set
`(47:34)` this exponent to zero
`(47:36)` that means you don't have this laplacian
`(47:38)` the second derivative here
`(47:42)` the other case is when you have
`(47:45)` set this to one here set this n to one
`(47:49)` then this will here be part of
`(47:52)` something like a diffusion term yeah
`(47:55)` this will go into a diffusion term
`(47:57)` and what you uh will then get is uh what
`(48:00)` is
`(48:00)` what what these kind of systems that
`(48:02)` describe are
`(48:04)` situations where the field supply is
`(48:06)` conserved
`(48:07)` now so you remove stuff at one point of
`(48:09)` the system if you remove stuff at one
`(48:11)` point of the system
`(48:12)` it has to pop up in another point
`(48:16)` an example is for example uh if you have
`(48:18)` something like hydrodynamics also where
`(48:20)` you just
`(48:20)` move mars around and uh but it doesn't
`(48:24)` really disappear
`(48:25)` and you just move things around but
`(48:27)` things don't disappear
`(48:28)` that's an example for these kind of
`(48:30)` model b
`(48:32)` systems yeah and
`(48:35)` uh so if you plug these things in so
`(48:37)` typical examples very famous example
`(48:39)` systems also called reaction diffusion
`(48:43)` systems
`(48:44)` now so and the typical reaction
`(48:45)` diffusion equation
`(48:47)` is seen here on the right hand side that
`(48:50)` describes a situation
`(48:52)` where uh the phi if you look here at the
`(48:55)` phi
`(48:56)` now then what happens what this term
`(48:59)` here does
`(49:00)` is a 5 is just a little bit larger than
`(49:03)` 0
`(49:04)` yeah then this term will be positive
`(49:07)` yeah and this this
`(49:10)` this will give to rise an increase in
`(49:12)` the field
`(49:14)` now this term is a diffusion term uh
`(49:17)` that just
`(49:17)` transports information to the set
`(49:19)` between different positions
`(49:21)` in the system and then we have our noise
`(49:24)` and this noise typically has
`(49:26)` prefactors it's multiplicative uh if you
`(49:29)` look for example at biological systems

slide 11

`(49:34)` so at the take home message for these
`(49:36)` spatial systems
`(49:38)` now we you can think about okay we just
`(49:40)` add an x variable well what i just said
`(49:42)` is just a complicated way of
`(49:44)` saying okay so we have some x variable
`(49:47)` and some diffusion yeah but everything
`(49:50)` else will look very similar as before
`(49:53)` now and indeed we can follow exactly the
`(49:56)` same steps now
`(49:57)` now we take the long-term equation
`(49:59)` general launch of my equation here
`(50:02)` you know which is this one i do exactly
`(50:05)` the same steps as before
`(50:07)` and now this here would be the first
`(50:10)` step
`(50:13)` now where we have this delta functions
`(50:16)` now now we don't only have a product
`(50:18)` over i
`(50:19)` that is already in the over time that is
`(50:21)` already in this delta function here
`(50:23)` but also over x yeah and then we just
`(50:26)` plug in this
`(50:27)` generalized launch of our equation and
`(50:29)` do the same step
`(50:31)` and then we get the field theory the
`(50:33)` martin citra rose functional
`(50:35)` for the spatially extended system so
`(50:38)` this looks pretty complicated
`(50:40)` but it is actually the same as before
`(50:43)` you know so we have we integrate again
`(50:46)` over the field fine and the response
`(50:50)` field
`(50:50)` file tilde and
`(50:54)` then we sum up different contributions
`(50:59)` to our variable to our observable like
`(51:02)` the second term
`(51:03)` and then as before we wait that
`(51:06)` with some exponential that essentially
`(51:10)` quantifies
`(51:10)` how far we are away from a realistic
`(51:14)` realization of the launching equation
`(51:17)` yeah and here we have
`(51:19)` exactly the same structure as before
`(51:22)` here you have the launch of the equation
`(51:25)` now you see you say that you need to
`(51:27)` solve the deterministic part of the
`(51:29)` moisture equation
`(51:31)` and this was previously just
`(51:34)` x tilde squared i said previously
`(51:40)` this was this term here
`(51:43)` was something like a
`(51:46)` over 2 x to the squared
`(51:50)` well now that looks more complicated
`(51:52)` yeah but it has the same form as if you
`(51:54)` look at this
`(51:56)` you have here your x tilde or phi tilde
`(52:00)` here you have another one that just
`(52:03)` coupled by the correlations in the noise
`(52:07)` uh that is described by this voice
`(52:10)` kernel
`(52:11)` now but the structure is the same as
`(52:14)` before
`(52:15)` yeah and uh so so uh so this is the
`(52:18)` martial citra rose
`(52:20)` functional for the noise for the
`(52:22)` spatially extended system
`(52:25)` yeah and with this uh i'm done with the
`(52:29)` definitions
`(52:29)` and with the actual
`(52:33)` derivation of the field theory now

slide 8

`(52:36)` let's see if we can apply it and now i
`(52:39)` go back
`(52:40)` to the example and just see how we're
`(52:43)` doing in time
`(52:47)` okay great i think almost exactly an
`(52:49)` hour
`(52:50)` so if you don't already know all of this
`(52:52)` maybe from last year's lecture
`(52:53)` then uh feel free to feel free for her
`(52:56)` to drop out
`(52:56)` and for the rest i'll go through one
`(52:58)` example
`(53:00)` and i'll double check so if i upload
`(53:02)` when i upload the uh
`(53:04)` the lecture notes i'll try to make sure
`(53:06)` that actually these minus signs
`(53:09)` are correct yeah so so if there was a
`(53:11)` mistake i'll correct it
`(53:12)` in the uploaded version that you find on
`(53:15)` the website
`(53:17)` excuse me yes so
`(53:20)` when you wrote the response function
`(53:23)` there was this i guess heavyside
`(53:28)` function multiplied with
`(53:29)` your autocorrelation so
`(53:33)` does that come up because uh you have
`(53:36)` a kind of the etho formalism
`(53:39)` into your system the point
`(53:44)` the response function if you have a
`(53:46)` perturbation here
`(53:48)` if you perturb at t prime and you look
`(53:50)` at the response at t
`(53:52)` now that's how this response function is
`(53:55)` defined
`(53:56)` answer you should have here that's a d
`(53:58)` prime
`(53:59)` here as you perturb at t prime and you
`(54:02)` look
`(54:03)` at the time t yeah then
`(54:06)` oh the other way around actually the
`(54:08)` answer so then then this
`(54:12)` herbicide function just ensures
`(54:15)` causality
`(54:16)` that you cannot observe the response
`(54:19)` that hap that happens before you could
`(54:20)` do the perturbation before you apply the
`(54:22)` force
`(54:25)` now so that's that's that's the reason
`(54:27)` why you get these tata functions
`(54:30)` okay okay thanks for the question
`(54:35)` let's go then to the
`(54:42)` first example
`(54:46)` here we go yeah and
`(54:49)` uh so let's let's have a look at the
`(54:52)` fields here for a very
`(54:54)` simple example and all like in the
`(54:56)` stochastic processes
`(54:58)` also simple examples quickly become
`(55:04)` complicated
`(55:06)` so let me find the notes
`(55:11)` here we go
`(55:18)` okay we start with a simple
`(55:22)` with probably one of the simplest larger
`(55:24)` way equation uh you can
`(55:26)` imagine and uh because it's so simple
`(55:29)` uh it has a name because it has been
`(55:32)` extensively
`(55:33)` studied and it's called the einstein
`(55:35)` ruling
`(55:36)` process and this process is just given
`(55:40)` by the time
`(55:41)` by the time derivative of x
`(55:45)` dot and this is equal to
`(55:48)` some restoring force all right so plus
`(55:51)` sometimes
`(55:52)` uh some external fluctuating force
`(55:56)` side and as always we have the usual
`(55:59)` conditions that are outside
`(56:00)` very nicely behaved doesn't have a mean
`(56:04)` and it's uncorrelated in time
`(56:08)` so now writing down the margin citra
`(56:12)` rose
`(56:13)` function integral easy because we just
`(56:16)` have to plug in this laundromate
`(56:18)` equation
`(56:19)` so martin
`(56:23)` martin cedar rose johnson the dominicus
`(56:28)` some people say martin said rose i did
`(56:30)` that once and i happened to be in munich
`(56:32)` and then i they were very angry that i
`(56:34)` forgot johnson
`(56:36)` my parents apparently had some
`(56:37)` connections to munich
`(56:40)` and uh now so that's the full
`(56:43)` that that that contains i think
`(56:45)` everybody who contributed
`(56:47)` uh
`(56:52)` reads now so what is the action now just
`(56:56)` i don't write the full integral just
`(56:57)` write the action
`(56:59)` so the action is
`(57:03)` dt i x tilde
`(57:12)` del t x plus alpha x
`(57:19)` now let's make sure that we fulfill the
`(57:21)` login equation
`(57:23)` plus a over 2
`(57:27)` dt
`(57:31)` x tilde now that would be squared
`(57:37)` now we can write down the generating
`(57:39)` function
`(57:41)` generating
`(57:47)` functional yeah and this is just as
`(57:50)` before
`(57:52)` that of h is tilde
`(57:57)` is equal to the integral
`(58:00)` over our two fields function interval of
`(58:02)` the word two fields
`(58:04)` field response field uh
`(58:07)` e to the minus s
`(58:11)` plus dt
`(58:15)` h x so that the external auxiliary field
`(58:19)` that comes to x
`(58:21)` plus an external field that couples
`(58:24)` to x tilde
`(58:28)` yeah and uh just
`(58:31)` just to make sure everybody understands
`(58:33)` now so why do i say that this field
`(58:35)` couples to x
`(58:36)` and x tilde here so these and why are
`(58:38)` these external fields
`(58:40)` provided like this this just follows if
`(58:42)` you write down
`(58:43)` um something like for example the
`(58:45)` ginsburg landlord theory also
`(58:48)` or you look at the icing model now then
`(58:50)` terms like this here
`(58:55)` will tilt the potential in one direction
`(58:59)` yeah and here this tilts the potential
`(59:04)` in the x-direction and this tills the
`(59:06)` potential in the axillary
`(59:07)` direction this tilts it in the x
`(59:10)` direction
`(59:12)` and this tilts it in the x tilde
`(59:14)` direction
`(59:16)` now so this is why we call these
`(59:17)` external fields
`(59:19)` and that they couple to uh
`(59:22)` these fields x and x that's the analogy
`(59:26)` to uh the ginsberg london theory and
`(59:29)` other fields
`(59:32)` okay so let's go this is the
`(59:35)` um functional generating functional
`(59:40)` and now we use that
`(59:44)` integral over dx e to the iq x
`(59:48)` is just delta of q so we use the
`(59:52)` definition of the fourier transform of
`(59:54)` the delta function
`(59:56)` and by doing that we get that
`(60:00)` this functional generative function
`(60:04)` is equal to d of
`(60:07)` x to the e
`(60:11)` to the a over two
`(60:16)` integral dt x to the squared
`(60:22)` plus dt
`(60:26)` our two fields h x
`(60:29)` plus h tilde x
`(60:33)` yeah and now we've got our delta
`(60:37)` function
`(60:39)` i del t minus alpha
`(60:44)` x to the plus h
`(60:53)` now when is this delta function here the
`(60:56)` delta function
`(60:58)` actually non-zero now so the delta
`(61:02)` delta functional
`(61:08)` is non zero
`(61:11)` if our x tilde
`(61:15)` solves this ordinary differential
`(61:18)` equation
`(61:20)` you know that we have here in the delta
`(61:22)` function now we can just write it down
`(61:26)` i times t to infinity
`(61:30)` dt prime e to the minus
`(61:35)` alpha t minus p prime
`(61:39)` h of t prime yeah
`(61:45)` now we substitute that so we saw that we
`(61:48)` already got rid of
`(61:49)` one field here by doing this reverse
`(61:53)` fury transform now we substitute that we
`(61:56)` get
`(61:57)` rid of our second field now we
`(61:59)` substitute
`(62:05)` into that
`(62:11)` and what we get
`(62:14)` is age h tilde
`(62:17)` is equal to and now comes a large
`(62:20)` exponential
`(62:21)` to see how to write that on the screen
`(62:26)` minus integral dt
`(62:30)` integral dt prime the second it comes
`(62:34)` from this x tilde
`(62:35)` and then we have
`(62:39)` a over 2 e to the minus
`(62:42)` alpha t minus t prime
`(62:48)` times h of t
`(62:53)` h of t prime plus
`(62:57)` i theta
`(63:00)` t minus t prime
`(63:03)` e to the minus alpha t
`(63:07)` minus t prime
`(63:10)` times i'm sorry i said it like
`(63:14)` even for the simplest process it gets
`(63:16)` lengthy
`(63:17)` h of t h of t prime
`(63:21)` yeah but what's happening here is
`(63:24)` nothing magical it's just a calculation
`(63:26)` such
`(63:26)` integrals well we substituted that x
`(63:29)` total
`(63:31)` into our generating functional and then
`(63:34)` we collected the terms
`(63:35)` and made sure
`(63:40)` that the right time order
`(63:43)` is given and now we have this generating
`(63:46)` function it looks a little bit
`(63:47)` complicated
`(63:48)` but we can deal with that now so we can
`(63:51)` take
`(63:52)` functional derivatives with this now for
`(63:54)` example to get a correlation function
`(64:02)` and so we have x
`(64:06)` of the x of t prime
`(64:10)` now we take the second
`(64:14)` derivative
`(64:19)` once at time t
`(64:24)` and once at time t prime
`(64:30)` and then we must not forget
`(64:33)` to set the fields equal to zero again
`(64:40)` now and if you look at this equation if
`(64:43)` we do that
`(64:44)` then and do a little bit of calculations
`(64:48)` don't
`(64:48)` get that actually this correlation
`(64:50)` function is
`(64:51)` a over alpha e to the minus alpha
`(64:56)` t minus t prime
`(65:02)` so that's just uh an example yeah so
`(65:05)` these actual calculations
`(65:06)` are a little bit messy but that's not
`(65:09)` just how you use these field theories
`(65:12)` that you have the field theory you write
`(65:14)` down the general functional
`(65:16)` yeah and then you look that you are able
`(65:19)` to
`(65:19)` take these functional derivatives with
`(65:22)` respect to these external fields
`(65:24)` and the rest is then say
`(65:28)` mathematics yeah
`(65:32)` okay great so this was an example and
`(65:35)` from next week
`(65:35)` we'll be a little bit more intuitive
`(65:37)` again now we've covered the technical
`(65:39)` stuff
`(65:40)` uh and we can look into some real uh
`(65:43)` physics
`(65:44)` and physics problems okay see you all
`(65:47)` next week
`(65:48)` bye