Generating-functional analysis of random Lotka–Volterra systems: A step-by-
step guide

Tobias Galla

Institute for Cross-Disciplinary Physics and Complex Systems IFISC (CSIC-UIB),

Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain

<EMAIL_ADDRESS>
## Abstract

This paper provides what is hopefully a self-contained set of notes describing
the detailed steps of a generating-functional analysis of systems of
generalised Lotka–Volterra equations with random interaction coefficients.
Nothing in these notes is original, instead the generating-functional method
(also known as the Martin-Siggia-Rose-DeDominic-Janssen formalism) and the
resulting dynamic mean field theories have been used for the study of
disordered systems and spin glasses for decades. But it is hard to find
unifying sources which would allow a beginner to learn step-by-step how these
methods can be used. My aim is to provide such a source. Most of the
calculations are specific to generalised Lotka–Volterra systems, but much can
be transferred to disordered systems in more general.

###### Contents

  1. 1 Introduction: What this is about, who might be interested in this, and how to use these notes
    1. 1.1 What this is about
    2. 1.2 Who might be interested in this and why?
    3. 1.3 How to use these notes
    4. 1.4 The notes are not a review
    5. 1.5 Structure of this document
  2. 2 Mean-field theory
    1. 2.1 Equilibrium mean field theory for the Ising model
    2. 2.2 Dynamic mean-field analysis for a kinetic Ising model
    3. 2.3 A word of caution
  3. 3 Mathematical tools: delta functions, Gaussian integrals and saddle-point method (skip if you are familiar with this)
    1. 3.1 Notation
    2. 3.2 Useful identities for delta functions
    3. 3.3 Fourier transforms of correlation functions
    4. 3.4 Gaussian integrals
    5. 3.5 Saddle-point integration (Laplace’s method)
      1. 3.5.1 Exponential function only
      2. 3.5.2 Product of exponentially dominated density and general function
  4. 4 The idea of generating functions and generating functionals
    1. 4.1 Generating functions for random variables
      1. 4.1.1 Scalar random variables
      2. 4.1.2 Vector-valued random variables
    2. 4.2 Generating functionals for stochastic processes
      1. 4.2.1 Discrete-time processes
      2. 4.2.2 Continuous-time stochastic processes
    3. 4.3 Correlation and response functions for discrete-time processes
      1. 4.3.1 Averages over realisations of the underlying stochastic process
      2. 4.3.2 Derivatives of the generating functional and correlation functions
      3. 4.3.3 Response functions
      4. 4.3.4 Moments of conjugate variables vanish
    4. 4.4 Continuous-time processes and functional derivatives
  5. 5 Toy example of a generating-functional calculation: Interacting processes without disorder
    1. 5.1 Setup
    2. 5.2 Heuristic derivation of the evolution equation for first moment in the thermodynamic limit
    3. 5.3 Generating-functional analysis I: Setup and saddle-point problem
      1. 5.3.1 Setting up the generating functional
      2. 5.3.2 Introduction of macroscopic order parameter
      3. 5.3.3 Formulation as a saddle-point problem
    4. 5.4 Generating functional analysis II: ‘Vanilla’ derivation of dynamic mean field process
      1. 5.4.1 Saddle-point equations for the order parameters
      2. 5.4.2 Elimination of hatted order parameter leads to single effective particle process
      3. 5.4.3 The problems with this simplified calculation
    5. 5.5 Generating functional analysis III: Proper derivation of dynamic mean-field process
      1. 5.5.1 Generating functional for the problem with additional perturbation fields
      2. 5.5.2 Physical meaning of the order parameter $M(t)$ at the saddle point
      3. 5.5.3 Proper derivation of the fact that $\widehat{M}(t)=0$ at the saddle point
      4. 5.5.4 Single effective particle measure and effective process
    6. 5.6 Summary of the main steps of the generating functional analysis
  6. 6 Generating functional analysis for random Lotka–Volterra equations I: Setup, disorder average and saddle-point equations
    1. 6.1 Overview of this section
    2. 6.2 Setup
      1. 6.2.1 Model definitions
      2. 6.2.2 Numerical generation of random matrix with correlations between diagonally opposed elements
    3. 6.3 Generating functional
    4. 6.4 Carrying out the disorder average
    5. 6.5 Introduction of macroscopic order parameters
    6. 6.6 Saddle-point analysis
      1. 6.6.1 Saddle-point equations for the macroscopic order parameters
      2. 6.6.2 Next steps
  7. 7 Generating functional analysis for random Lotka–Volterra equations II: ‘Vanilla’ derivation of the effective single-species process
    1. 7.1 Simplification of saddle-point equations
    2. 7.2 Interpretation in terms of an effective single-species process
  8. 8 Generating functional analysis for random Lotka–Volterra equations III: Proper derivation of single effective-species process
    1. 8.1 Identification of order parameters
      1. 8.1.1 Step 1: Definition of order parameters from the original Lotka–Volterra problem
      2. 8.1.2 Step 2: Order parameters for the original problem as derivatives of the disorder-averaged generating functional
      3. 8.1.3 Step 3: Equivalence to order parameters at the saddle point
    2. 8.2 Effective process for a representative species
    3. 8.3 Comments on the legitimacy of the vanilla approach
  9. 9 Generating functional analysis for random Lotka–Volterra equations IV: Dynamic mean-field process and its interpretation
    1. 9.1 Mathematical structure
    2. 9.2 Interpretation
    3. 9.3 Summary so far
  10. 10 Fixed point analysis
    1. 10.1 Fixed-point ansatz and its consequences
      1. 10.1.1 Fixed-point ansatz for effective abundances
      2. 10.1.2 The noise variable in the effective process becomes static
      3. 10.1.3 Distribution of fixed-point abundances
    2. 10.2 Self-consistency relations for order parameters
      1. 10.2.1 Summary of status so far
      2. 10.2.2 Self-consistent expressions for order parameters $M^{*}$ and $q$
      3. 10.2.3 Self-consistent relation for the integrated response $\chi$
      4. 10.2.4 Summary of self-consistency relations at the fixed point
    3. 10.3 Further analysis
    4. 10.4 Solution procedure
    5. 10.5 Interpretation of the order parameters
    6. 10.6 Species abundance distribution
    7. 10.7 Test against numerical integration of the Lotka–Volterra dynamics
  11. 11 Stability analysis and phase diagram
    1. 11.1 Onset of divergent solutions
    2. 11.2 Linear stability analysis
      1. 11.2.1 Step 1: Add small perturbation to effective single-species dynamics
      2. 11.2.2 Step 2: Independence of random variables
      3. 11.2.3 Step 3: Further analysis for species with vanishing fixed-point abundance
      4. 11.2.4 Step 4: Further analysis for non-zero fixed-point abundances
      5. 11.2.5 Step 5: Self-consistency relations
      6. 11.2.6 Step 6: Criterion for linear instability
    3. 11.3 Stability diagram
  12. 12 Cavity approach to dynamic mean-field theory
  13. 13 Summary and discussion
  14. A Derivation of the effective process for the Lotka–Volterra problem without division by species abundances
  15. B Equivalence of averages over the effective dynamics and averages in the original Lotka–Volterra problem

##  1 Introduction: What this is about, who might be interested in this, and
how to use these notes

###  1.1 What this is about

In these notes I describe how so-called ‘generating functionals’ can be used
to study generalised Lotka-Volterra dynamics with random interaction matrices.
These equations describe the time-evolution of non-negative abundances
$x_{1},\dots,x_{N}$ of $N$ species, and are of the form [1, 2, 3]

$\dot{x}_{i}=x_{i}\left(1-x_{i}+\sum_{j}\alpha_{ij}x_{j}\right),$ (1)

where $i=1,\dots,N$, and where the $\alpha_{ij}$ are ‘quenched’ random
variables. This means that the $\alpha_{ij}$ are drawn at the beginning from a
joint probability distribution (to be specified). Then they are kept fixed,
and we are interested in the behaviour of the solution of the coupled
differential equations in (1), i.e., of the resulting $x_{i}(t)$. The initial
condition for the $x_{i}(t=0)>0$ can also be random, i.e. , the
$x_{1}(0),\dots,x_{N}(0)$ are (or can be) drawn from some joint probability
distribution. Crucially however, the generalised random Lotka-Volterra
dynamics contains no further stochasticity during the time-evolution of the
system.

The initial Lotka–Volterra model only described two species (see e.g. [4]).
The term ‘generalised’ is commonly used to describe generalisations to systems
with more than two species. Throughout this document we will be looking at
systems with a large number $N$ of species (technically, we will be
considering the thermodynamic limit $N\to\infty$), but we will often omit the
word ‘generalised’. We also have random interaction coefficients, and we will
simply call this the random Lotka–Volterra model, or the disordered
Lotka–Volterra equations. Sometimes we will use combinations of these terms.

Given that Eqs. (1) are nonlinear, it is hopeless to expect analytical
progress for any particular realisation of the $\alpha_{ij}$. However, and as
we will see, the typical behaviour of the dynamics can be characterised, at
least in the thermodynamic limit. To do this one derives what is called a
‘dynamical mean field theory’, capturing what a typical ‘representative’
species experiences. The purpose of these notes is to describe how to do this,
and how to analyse the resulting mean field dynamics.

The methods we will use, generating functionals and dynamic mean-field theory,
are well established in the theory of disordered systems. There are plenty of
sources, see for example [5, 6, 7, 8, 9, 10, 11, 12]. The application of these
methods to random Lotka–Volterra equations or related replicator equations is
not new either, see [13, 14, 3, 15, 16], and in a sense these notes are an
expanded variant of the Supplementary Material of [3]. More recent work
includes [17, 18, 19, 20, 21].

Note: This version is not the final version. Please send me comments and
corrections.
The document you are looking at on your screen (or on paper) is not the final
version of these notes. I have done a fair amount of checking, and a number of
my students have pointed out a number of typos (and I have corrected those).
But inevitably there will still be factors of $i$ or $2\pi$ missing on
occasion. This is the version I decided to put on the ArXiv for the time
being. If anyone really reads this and finds mistakes, please email me with
corrections. If you think the text could be made clearer somewhere, let me
know.

###  1.2 Who might be interested in this and why?

An earlier version of these notes was initially written as an internal
document for masters and PhD students who were trying get into the topic. This
remains the natural primary audience, along with other non-expert researchers.
I hope that what I have written is sufficiently didactic and self-contained.
There might also be readers who are already familiar with tools for disordered
systems, and who’d like to apply these methods to models of complex
ecosystems. And then there are perhaps readers who are already experts, but
who’d like to look up particular technical details.

###  1.3 How to use these notes

Readers who are experienced with generating-functional methods for disordered
systems will not need any advice on how to use the notes. They know what they
are doing. They will probably not read through this paper line-by-line.
Instead they’ll just skim through the first pages, recognise most equations,
mumble something to the effect of ‘ahh, yes, this here is this, and this here
is that’, turning the pages quite quickly. Then they might look at a few
technical points. Perhaps they’ll identify some of sort of subtlety that I
didn’t see. If you are such a reader and find anything that is wrong,
incomplete, or requires further discussion, please let me know. If you think
the presentation is unnecessarily complicated and that things could be
explained more efficiently, please also get in touch.

Things could look a little more daunting for someone who is new to all of
this. There are so many quantities and symbols here, individual steps can be
quite technical, and the whole thing is just very long. It is hard to see the
forest for the trees. There are many little things to get hung up on. A
beginner might read the notes line-by-line, diligently going from one equation
to the next, and they will then (most likely) get stuck at some point. It’ll
then probably take a bit of courage to proceed.

If these methods are all new to you, then my advice is that you initially try
to go through the notes line by line. But if you get stuck somewhere, don’t
spend hours and hours thinking about one little detail. Proceed. As a rule of
thumb, if it takes you more than 5-10 minutes to understand a step, just park
it, and keep going. You can return to it later. There are so many subtle
points here, and you can’t worry about everything from the beginning. Adapting
from ACC Coolen et al’s draft book manuscript [22], if you build a house you
first construct the foundations and decide how many rooms you want, and what
room is next to what other room etc. Then you build the main walls, and
perhaps the roof. You worry about the colour of the tiles in the bathroom and
the shape of the door handles at the end. It is the same here, first try to
get an idea of the general structure of the calculation, the main steps.
That’s figuring out how many rooms the house has, and where the doors are and
the staircase. You worry about the many small and subtle issues later.

I tried to write these notes to make this as easy as possible. Occasionally
there are steps in the calculation where too much worry about overly technical
points would be distracting. On the other hand, I wanted to be as precise as I
could. Occasionally, I had to be slightly sloppy to keep the story straight.
But I’ll tell you when I do this (or more accurately, when I am aware that I
am doing this). I then discuss the finer points later, for example in an
appendix. Having said this, there will inevitably be subtleties that I missed.
If you find any, please let me know.

###  1.4 The notes are not a review

I am not aware of a review of the large (and growing) body of work by
statistical physicists on random Lotka–Volterra dynamics or related systems.
Perhaps someone will write one at some point. These notes are definitely not
such a review. This is not the intention. Instead, I am giving a relatively
detailed account of the necessary calculations. I only focus on the most basic
random Lotka–Volterra model. A number of variations and extensions have been
studied with generating-functional methods or other techniques from the theory
of disordered systems, see for example [2, 23, 15, 16, 24, 25, 17]. I hope
that the interested reader will be able to adapt the methods once they have
worked through these notes. The notes end at the point where the phase diagram
has been established, but of course much more is known about the model [2, 23,
16, 24, 25, 17, 26, 27]. Again, this is not included here, but I hope that
studying the present notes will make it easier to read the more advanced
literature.

I am also conscious that I am not really motivating why it is interesting to
study Lotka–Volterra equations with random interactions. I assume that anyone
who is reading this is already motivated. If you are not, then you can find
some more background in the references above. More widely, the study of random
ecosystems is central to the so-called ‘stability-complexity’ debate in
theoretical ecology. There is plenty of material on this, see for example [28,
29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. I do not include
details here, to keep the notes focused on the main purpose, to introduce the
reader to the mathematical techniques.

###  1.5 Structure of this document

These notes can be divided into three parts:

Part I:

  * •

Section 1: General introduction;

  * •

Section 2: Brief background on ‘mean-field theory’;

  * •

Section 3: Collection of the main mathematical tools we will be using in the
generating-functional analysis.

Sections 2 and 3 are included for completeness, and can be skipped by readers
who are familiar with these concepts.

Part II:
This part intended as an introduction for readers who are not familiar with
the idea of a generating function for probability distributions, and/or
generating functionals for stochastic processes.

  * •

Section 4: The idea of generating functions and functionals for probability
distributions, and simple stochastic processes;

  * •

Section 5: Application of generating functionals to a high-dimensional toy
model, with no disorder.

Part III:
This part contains the actual generating-functional analysis for the
Lotka–Volterra model.

  * •

Section 6: We set up the generating functional for the disordered
Lotka–Volterra model, carry out the disorder average, introduce the relevant
macroscopic dynamic order parameters, and derive the saddle-point equations
for these order parameters in the limit of an infinite number of species;

  * •

Section 7: We first present a simplified (‘vanilla’) derivation of the
corresponding dynamic mean field theory from the saddle-point equations. This
is based on certain assertions, which can only be justified retrospectively.

  * •

Section 8: This section contains are more complete bottom-up derivation of the
same effective dynamics as in Sec. 7. We also relate the order parameters at
the saddle point to those in the original Lotka–Volterra problem.

  * •

Section 9: We discuss the mathematical structure and the interpretation of the
dynamic mean field theory for the Lotka–Volterra model. This theory consists
of the effective single-species process, and the self-consistency relations
for the order parameters that complement the representative-species dynamics.

  * •

Section 10: Here, we make a fixed-point ansatz for the effective process, and
derive the self-consistency relations for the corresponding static order
parameters. We describe how these can be solved parametrically in closed form.

  * •

Section 11: We study the stability of the fixed point from Sec. 10, and
establish a phase (or stability) diagram for the disordered generalised
Lotka–Volterra system.

  * •

Section 12: In this supplementary section we briefly describe an alternative
derivation of the effective dynamics using the so-called ‘cavity’ approach.
The section is not required for the analysis in Secs. 6-11, and is included
mostly for completeness.

Parts I-III are complemented by a summary and discussion in Sec. 13, and by
two appendices in which we provide supplementary details for some parts of the
calculation in Part III.

##  2 Mean-field theory

###  2.1 Equilibrium mean field theory for the Ising model

In most textbooks or lecture notes on statistical physics mean-field theory is
introduced in the context of the Ising model (Refs. [43, 44] are examples of
good sources). We’ll briefly summarise this here.

A typical model describes $N$ spins, $s_{i}$, sitting on some sort of network
(usually a regular lattice). For simplicity, we assume a regular network,
i.e., each spin has the same number of nearest neighbours. We write $k$ for
this coordination number.

The spins can each be up or down, $s_{i}=\pm 1$. We write
$\mathbf{s}=(s_{1},\dots,s_{N})$ for a spin configuration. The total energy of
the system is given by

$H(\mathbf{s})=-h\sum_{i}s_{i}-J\sum_{i}s_{i}\sum_{j\in i}s_{j}.$ (2)

The notation $j\in i$ describes the nearest neighbours of spin $i$. The
quantity $J$ is a coupling strength, and $h$ is an external field. Without
loss of generality we set $J=1$.

The idea of mean field theory is to replace the sum $\sum_{j\in i}s_{j}$ by a
‘mean field’ acting on any one spin. More precisely, we replace

$\sum_{j\in i}s_{j}\to km,$ (3)

where $m$ is the mean magnetisation of the spins neighbouring $i$, i.e.,

$m=\frac{1}{k}\sum_{j\in i}\left\langle{s_{j}}\right\rangle.$ (4)

This quantity is taken to be the same for all spins $i$. Just to be clear, the
word ‘mean’ describes an average over the Boltzmann distribution for the
model111When you see (or use) angle brackets
$\left\langle{\dots}\right\rangle$ it is always a good idea to ask yourself
what type of average exactly these brackets represent.. We will come to this
in a minute.

We can then write

$H=-h_{mf}\sum_{i}s_{i},$ (5)

with the ‘mean field’

$h_{mf}=h+km.$ (6)

Within this approximation and using the usual Boltzmann distribution at
inverse temperature $\beta$, the probability for spin $i$ to be in state
$s_{i}=\pm 1$ is proportional to $e^{-\beta h_{mf}s_{i}}$. Therefore we have

$P(\mbox{spin $i$ is in state $s_{i}=\pm 1$})=\frac{e^{-\beta
h_{mf}s_{i}}}{2\cosh(\beta h_{mf})}.$ (7)

It remains to determine the mean field self-consistently. Eq. (7) implies that
$\left\langle{s_{i}}\right\rangle$ is the same for all $i$, and this value
must then be the same as the magnetisation $m$ in Eq. (4). Therefore, we have

$\displaystyle m$ $\displaystyle=$ $\displaystyle P(\mbox{spin $i$ is
up})-P(\mbox{spin $i$ is down})$ (8) $\displaystyle=$
$\displaystyle\tanh(\beta h_{mf})$ $\displaystyle=$
$\displaystyle\tanh[\beta(h+km)].$

So, the result from this analysis is that the magnetisation of the model as a
function of temperature, the mean coordination number $k$, and the external
field $h$, is obtained as the solution of the equation

$m=\tanh[\beta(h+km)].$ (9)

This can then be analysed to identify paramagnetic and ferromagnetic phases,
the transition between the two and so on. We won’t go into detail here. Our
main point is that we have obtained the self-consistent equation (9) for the
macroscopic order parameter $m$. The word ‘macroscopic’ indicates that $m$ is
a property of the system as a whole, not a spin-specific quantity ($m$ carries
no site index $i$).

We note that this was a static equilibrium analysis. We have not said anything
about time-evolution, nothing in our analysis was ‘dynamic’. Instead we
assumed a Boltzmann distribution at equilibrium. The order parameter $m$ is
not time dependent, it is a static macroscopic order parameter.

###  2.2 Dynamic mean-field analysis for a kinetic Ising model

We’ll now look at the Ising model again, but from a dynamic point of view.
This means that we start from an actual dynamic process governing the
evolution of the spins in time. Of course, this will be closely related to the
static picture in the previous section, but I hope you see the difference.

Say the spins at time $t$ are in the configuration
$\mathbf{s}(t)=[s_{1}(t),\dots,s_{N}(t)]$. We then assume that, in the next
time step, each spin is updated according to the following rule,

$P[s_{i}(t+1)=\pm 1]=\frac{e^{\mp\beta h_{i}[\mathbf{s}(t)]}}{2\cosh[\beta
h_{i}[\mathbf{s}(t)]},$ (10)

where $h_{i}[\mathbf{s}(t)]$ is the total field acting on spin $i$ at that
time,

$h_{i}[\mathbf{s}(t)]=h+\sum_{j\in i}s_{j}(t).$ (11)

We now make a similar approximation as in the static case, and introduce

$h_{mf}[m(t)]=h+km(t),$ (12)

where

$m(t)=\frac{1}{k}\sum_{j\in i}\left\langle{s_{j}(t)}\right\rangle.$ (13)

We again assume that the quantity on the right does not depend on the site
index $i$. Inserting into Eq. (10), and averaging, we then arrive at

$\left\langle{s_{i}(t+1)}\right\rangle=\tanh[\beta(h+km(t))],$ (14)

for all $i$. The average on the left-hand side is $m(t+1)$, and we have
therefore arrived an the following equation for the dynamical macroscopic
order parameter $m(t)$,

$m(t+1)=\tanh[\beta(h+km(t))].$ (15)

The fixed points of this dynamics fulfill Eq. (9), but Eq. (15) delivers more
than that, it describes the dynamical evolution of the magnetisation in time
(within the mean-field approximation).

###  2.3 A word of caution

Our main focus is on the generalised Lotka-Volterra equations,

$\dot{x}_{i}=x_{i}\left(1-x_{i}+\sum_{j}\alpha_{ij}x_{j}\right),$ (16)

with quenched random coefficients $\alpha_{ij}$. Say the distribution of the
$\alpha_{ij}$ is such that the mean of each $\alpha_{ij}$ (for given $i$ and
$j$) is $\mu/N$. One might then be tempted to think that a mean-field approach
could look something like this:

$\dot{m}(t)=m(t)\left[1-m(t)+\mu m(t)\right],$ (17)

where $m(t)=\overline{x_{i}(t)}$ (we generall use an overbar
$\overline{\cdots}$ for the average over the disorder, i.e., the random
couplings $\alpha_{ij}$). Unfortunately, things are not as simple as this – if
they were, the variance of the $\alpha_{ij}$ would be irrelevant, and a model
in which all $\alpha_{ij}$ are the same would lead to the same mean-field
dynamics as a truly disordered model. That’s not the case. For example, if
$\mu=0$, the random Lotka–Volterra system will not just converge to $m(t)\to
1$ if started from positive $x_{i}(t=0)$. Instead, as we will see later, there
is a transition between a stable regime for small variance of the interaction
coefficients and an unstable regime for larger variances. This means that
we’ll need to work a little harder to derive a dynamical mean-field theory for
the random Lotka–Volterra model.

##  3 Mathematical tools: delta functions, Gaussian integrals and saddle-point
method (skip if you are familiar with this)

Before we can start with our mean-field analysis of the random Lotka–Volterra
equations, we have to introduce some notation, and a few general concepts.

###  3.1 Notation

Throughout the paper, bold-face letters indicate column vectors, e.g.
$\mathbf{x}=(x_{1},\dots,x_{N})^{T}$. The superscript $T$ stands for
‘transposed’. Vector components are indicated by latin indices, for example
$x_{i}$ is the $i$-th component of $\mathbf{x}$.

Underlined symbols represent scalar functions of time. If time is discrete, we
write for example $\underline{x}=(x(t=0),x(t=1),\dots)^{T}$. Occasionally
these objects are also treated as vectors (for example to be multiplied by a
matrix). We understand these to be column vectors, hence the transpose sign.
We will also use this notation when time is continuous. Time is indicated as
an argument in brackets, for example $x(t)$.

Multi-component functions of time are written as underlined bold-face symbols.
For example, $\underline{\mathbf{x}}$ stands for
$\underline{x}_{1},\underline{x}_{2},\dots,\underline{x}_{N}$, where each of
the $\underline{x}_{i}$ is a function of time.

Matrices in ‘time space’ are indicated by double underline, e.g.
$\underline{\underline{C}}$, with entries $C(t,t^{\prime})$.

Matrices in component space are denoted by bold face capital letters, e.g.
$\mathbf{A}$ has elements $a_{ij}$, where $i,j=1,\dots,N$.

We use the following convention for Fourier transforms and the inverse
transform:

$\displaystyle\widetilde{y}(\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}}\int dt\,e^{-i\omega t}y(t),$
$\displaystyle y(t)$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2\pi}}\int
dt\,e^{i\omega t}\widetilde{y}(t).$ (18)

###  3.2 Useful identities for delta functions

First, we recall the definition of the Dirac delta function,

$\int_{a}^{b}dx~{}f(x)\delta(x-x_{0})=f(x_{0}),$ (19)

if $x_{0}\in[a,b]$. Otherwise the integral is zero. The delta function can be
written in its so-called ‘exponential representation’. This can be derived by
carrying out a Fourier transform of $\delta(\cdot)$:

${\cal
F}[\delta](\widehat{x})=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dx~{}\delta(x)~{}e^{-i\widehat{x}x}=\frac{1}{\sqrt{2\pi}}.$
(20)

In the last step we have used Eq. (19). We have written ${\cal F}$ for the
Fourier transform, and we have used $\widehat{x}$ as the variable conjugate to
$x$. Eq. (20) means that the Fourier transform of the $\delta$-function is
flat (independent of $\widehat{x}$).

Applying the reverse Fourier transform we then have

$\delta(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\widehat{x}~{}e^{i\widehat{x}x}.$
(21)

Using $1=\int_{-\infty}^{\infty}dx~{}\delta(x)$, we therefore have

$1=\int_{-\infty}^{\infty}\frac{dxd\widehat{x}}{2\pi}~{}e^{i\widehat{x}x}$
(22)

We also have $1=\int_{-\infty}^{\infty}dx\,\delta(x-a)=1$, for any constant
$a$ (i.e., an $a$ which does not depend on $x$). Therefore

$1=\int_{-\infty}^{\infty}\frac{dxd\widehat{x}}{2\pi}~{}e^{i\widehat{x}(x-a)}.$
(23)

###  3.3 Fourier transforms of correlation functions

Consider a real-valued stochastic process $y(t)$ in continuous time, with zero
average, i.e., $\left\langle{y(t)}\right\rangle=0$ for all $t$. We define the
correlation function

$C_{y}(t,t^{\prime})=\left\langle{y(t)y(t^{\prime})}\right\rangle,$ (24)

where the angle bracket stands for an average over realisations of the
process. Assuming that the process is such that it reaches a stationary state
eventually, we have time-translation invariance at long times, i.e.,
$\lim_{t\to\infty}C_{y}(t+\tau,t)$ is a function of time difference $\tau$
only. We write

$C(\tau)=\lim_{t\to\infty}C(t+\tau,t).$ (25)

With the definition

$\widetilde{y}(\omega)=\frac{1}{\sqrt{2\pi}}\int dt\,e^{-i\omega t}y(t)$ (26)

for the Fourier transform of $y(\cdot)$. Using the fact that $y(t^{\prime})$
is real valued, and writing $t^{\prime}=t-\tau$, we then have

$\displaystyle\left\langle{\widetilde{y}(\omega^{\prime})^{*}\widetilde{y}(\omega)}\right\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi}\int
dtdt^{\prime}\,e^{i(\omega^{\prime}t^{\prime}-\omega
t)}\left\langle{y(t)y(t^{\prime})}\right\rangle$ (27) $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int
dtd\tau\,e^{-i\omega\tau}e^{i(\omega^{\prime}-\omega)t}\left\langle{y(t)y(t-\tau)}\right\rangle$
$\displaystyle=$ $\displaystyle\delta(\omega-\omega^{\prime})\int d\tau
C(\tau)e^{-i\omega^{\prime}\tau}$ $\displaystyle=$
$\displaystyle\sqrt{2\pi}\delta(\omega-\omega^{\prime})\widetilde{C}(\omega).$

We have used the fact that
$\left\langle{y(t)y(t+\tau)}\right\rangle=\left\langle{y(t)y(t-\tau)}\right\rangle$
in the stationary regime.

This result is often summarised as follows:
$\left\langle{|\widetilde{y}(\omega)|^{2}}\right\rangle$ is the Fourier
transform of $C(\tau)$. There are two issues here. One is the pre-factor
$\sqrt{2\pi}$, that’s not a problem ‘is the Fourier transform’ simply means
’is the Fourier transform up to trivial pre-factors’.

The other issue is more subtle, namely, the fact that the left-hand side of
Eq. (27) contains two frequencies, $\omega$ and $\omega^{\prime}$, and the
right-hand side the delta function $\delta(\omega-\omega^{\prime})$.
Nonetheless it is common to use abbreviated statements of the form
$\widetilde{C}(\omega)=(2\pi)^{-1/2}\left\langle{|\widetilde{y}(\omega)|^{2}}\right\rangle$.

If $y(t)$ is white noise of unit variance, i.e.,
$\left\langle{y(t)y(t^{\prime})}\right\rangle=\delta(t-t^{\prime})$, we have
$C(\tau)=\delta(\tau)$, and therefore
$\widetilde{C}(\omega)\equiv\frac{1}{\sqrt{2\pi}}$ for all $\omega$ (a flat
spectrum is why this is called white noise after all). It then follows that

$\left\langle{\widetilde{y}(\omega^{\prime})^{*}\widetilde{y}(\omega)}\right\rangle=\delta(\omega-\omega^{\prime}).$
(28)

Often this is simply written as
$\left\langle{|\widetilde{y}(\omega)|^{2}}\right\rangle=1$.

We mention a third subtlety. In the third step in Eq. (27) we have made the
replacement $\left\langle{y(t)y(t-\tau)}\right\rangle\to C(\tau)$. This is
valid only in the stationary state, i.e., for large times $t$. But $t$ is an
integration variable in Eq. (27). There are two ways of interpreting this. One
can implicitly assume that the starting point of the dynamics is at
$t\to-\infty$. Any finite time $t$ is then automatically in the stationary
state. Alternatively one can think of starting the dynamics at $t=0$ but from
initial conditions drawn from the stationary distribution of the process.

###  3.4 Gaussian integrals

We will need various multivariate Gaussian integrals, in particular ones of
the form

$\int_{-\infty}^{\infty}dx_{1}\cdots
dx_{N}\exp\left(-\frac{1}{2}\sum_{i,j=1}^{N}x_{i}A_{ij}x_{j}+\sum_{i=1}^{N}b_{i}x_{i}\right)\equiv\int
d\mathbf{x}~{}\exp\left(-\frac{1}{2}\mathbf{x}^{T}\mathbf{A}\mathbf{x}+\mathbf{b}^{T}\mathbf{x}\right).$
(29)

The $N\times N$ matrix $\mathbf{A}$ is assumed to be symmetric and positive
definite.

The integral can be carried out (for example by transforming to the space
spanned by the eigenvectors of $\mathbf{A}$), and one has

$\int
d\mathbf{x}~{}\exp\left(-\frac{1}{2}\mathbf{x}^{T}\mathbf{A}\mathbf{x}+\mathbf{b}^{T}\mathbf{x}\right)=\sqrt{\frac{(2\pi)^{N}}{\det\mathbf{A}}}\exp\left(\frac{1}{2}\mathbf{b}^{T}\mathbf{A}^{-1}\mathbf{b}\right).$
(30)

In the case $N=1$, and setting $\mathbf{A}=\frac{1}{\sigma^{2}}$ this reduces
to the well-known identity

$\frac{1}{\sqrt{2\pi\sigma^{2}}}\int
dx\exp\left(-\frac{x^{2}}{2\sigma^{2}}+bx\right)=\exp\left(\frac{1}{2}\sigma^{2}b^{2}\right).$
(31)

Multivariate Gaussian distribution:
We also note that

$p_{\mathbf{C}}(\mathbf{x})=(2\pi)^{-N/2}(\det\mathbf{C})^{-1/2}\exp\left(-\frac{1}{2}\mathbf{x}^{T}\mathbf{C}^{-1}\mathbf{x}\right)$
(32)

is a normalised probability distribution, for a symmetric and positive
definite $N\times N$ matrix, $\mathbf{C}$. In other words $\int
d\mathbf{x}~{}p_{\mathbf{C}}(\mathbf{x})=1$. This follows from Eq. (30), with
the replacement $\mathbf{A}\to\mathbf{C}^{-1}$, and $\mathbf{b}=\mathbf{0}$.

We also have

$\int d\mathbf{x}~{}p_{\mathbf{C}}(\mathbf{x})x_{i}x_{j}=C_{ij}.$ (33)

This can be derived from Eq. (30) by taking derivatives with respect to
$b_{i}$ and $b_{j}$, evaluated at $\mathbf{b}=\mathbf{0}$.

Equation (33) indicates that the matrix $\mathbf{C}$ is the correlation matrix
of the Gaussian random variables $x_{i}$, $i=1,\dots,N$. This also justifies
why we assume that the matrix $\mathbf{A}=\mathbf{C}^{-1}$ was symmetric and
positive definite. Eq. (33) shows that $\mathbf{C}$ is symmetric, and this in
turn means that $\mathbf{C}^{-1}$ is symmetric. Further we have for any (non-
zero) vector $\mathbf{v}$ of length $N$,

$\displaystyle\mathbf{v}^{T}\mathbf{C}\mathbf{v}$ $\displaystyle=$
$\displaystyle\sum_{ij}v_{i}v_{j}\int
d\mathbf{x}~{}p_{\mathbf{C}}(\mathbf{x})x_{i}x_{j}$ (34) $\displaystyle=$
$\displaystyle\int
d\mathbf{x}~{}p_{\mathbf{C}}(\mathbf{x})\left(\sum_{k}v_{k}x_{k}\right)^{2}>0.$

Hence $\mathbf{C}$ is positive definite.

Continuous Gaussian distribution:
Equation (30) can directly be transferred to the time domain:

$\int
d\underline{x}~{}\exp\left(-\frac{1}{2}\underline{x}^{T}\underline{\underline{A}}\,\underline{x}+\underline{b}^{T}\underline{x}\right)=\sqrt{\frac{(2\pi)^{N}}{\det\underline{\underline{A}}}}\exp\left(\frac{1}{2}\underline{b}^{T}\underline{\underline{A}}^{-1}\underline{b}\right).$
(35)

If there is no endpoint in time ($t=0,1,2,\dots$ unbounded), then the vectors
and matrices are infinite dimensional. The space of $\underline{x}$ is then an
infinite-dimensional vector space.

The formula in Eq. (35) can also be used when the time index is continuous. We
then have $\underline{b}^{T}\underline{x}=\int dt~{}b(t)x(t)$ for the scalar
product, and the object
$\underline{x}^{T}\underline{\underline{A}}\,\underline{x}$ is given by

$\underline{x}^{T}\underline{\underline{A}}\,\underline{x}=\int
dt~{}dt^{\prime}~{}x(t)A(t,t^{\prime})x(t^{\prime}).$ (36)

The object $\underline{x}$ is now an element of a space of functions with one
continuous argument $t$, and $\underline{\underline{A}}$ is a linear operator,
mapping one such function onto another.

###  3.5 Saddle-point integration (Laplace’s method)

In this section we briefly discuss Laplace’s method to carry out integrals of
the form $\int dxg(x)\exp[Nf(x)]$ in the limit $N\to\infty$. Physicists
sometimes also refer to this as ‘saddle-point integration’. In mathematics,
saddle-point integration or ‘the method of steepest descent’ refers to wider
set of integration techniques in the complex plane [45, 46]. A concise summary
can also be found in [47].

####  3.5.1 Exponential function only

Suppose we are faced with an integral of the type

$I\equiv\int dxe^{Nf(x)},$ (37)

and we are interested in the behaviour for $N\gg 1$. We also assume that
$f(\cdot)$ has a unique global maximum at $x_{0}$. For large $N$ the integral
$I$ will therefore be dominated by the region around $x_{0}$. We can then
expand

$f(x)=f(x_{0})+\frac{1}{2}f^{\prime\prime}(x_{0})(x-x_{0})^{2}+\cdots.$ (38)

We note that $f^{\prime}(x_{0})=0$ and $f^{\prime\prime}(x_{0})<0$ (because
$f$ has a local maximum at $x_{0}$). We then have

$\displaystyle I$ $\displaystyle=$ $\displaystyle\int
dxe^{Nf(x_{0})+N\frac{1}{2}f^{\prime\prime}(x_{0})(x-x_{0})^{2}+\cdots}$ (39)
$\displaystyle=$ $\displaystyle e^{Nf(x_{0})}\int
dxe^{N\frac{1}{2}f^{\prime\prime}(x_{0})(x-x_{0})^{2}+\cdots},$

and we can proceed by Gaussian integration. We find

$I\approx\sqrt{\frac{2\pi}{N|f^{\prime\prime}(x_{0})|}}e^{Nf(x_{0})}.$ (40)

####  3.5.2 Product of exponentially dominated density and general function

Suppose now, we know that $p(x)$ is normalised, $\int dx~{}p(x)=1$, and of the
form $p(x)=Ae^{Nf(x)}$, with $f$ as above. Then we must have
$A=\sqrt{\frac{|f^{\prime\prime}(x_{0})|N}{2\pi}}e^{-Nf(x_{0})}$.

We now follow for example [48] and write $x=x_{0}+\frac{y}{\sqrt{N}}$, and
think of things as an expansion in $N^{-1/2}$ (as opposed to in $x-x_{0}$). We
then have

$f(x)=f(x_{0})+\frac{1}{2N}y^{2}f^{\prime\prime}(x_{0})+{\cal O}(N^{-3/2}),$
(41)

i.e.

$Nf(x)=Nf(x_{0})+\frac{1}{2}y^{2}f^{\prime\prime}(x_{0})+{\cal O}(N^{-1/2}).$
(42)

This means

$e^{Nf(x)}=e^{Nf(x_{0})}e^{\frac{1}{2}y^{2}f^{\prime\prime}(x_{0})}\left(1+{\cal
O}(N^{-1/2})\right).$ (43)

For a function $g(x)$ we also have

$g(x)=g(x_{0})\left(1+\frac{yg^{\prime}(x_{0})}{g(x_{0})\sqrt{N}}+{\cal
O}(N^{-1})\right).$ (44)

Using $p(x)=Ae^{Nf(x)}$, with
$A=\sqrt{\frac{|f^{\prime\prime}(x_{0})|N}{2\pi}}e^{-Nf(x_{0})}$as given
above, we then find to leading order in powers of $N^{-1/2}$,

$\int
dx\,p(x)g(x)\approx\sqrt{\frac{|f^{\prime\prime}(x_{0})|}{2\pi}}g(x_{0})\int
dy\,e^{\frac{1}{2}y^{2}f^{\prime\prime}(x_{0})},$ (45)

where we have shifted the integration variable from $x$ to $y$ (using
$dx=dy/\sqrt{N}$). We then carry out the remaining Gaussian integral over $y$
(using the fact that $f^{\prime\prime}(x_{0})<0$), and find

$\int dx\,p(x)g(x)\approx g(x_{0}).$ (46)

Bottom line: If we are doing averages against a normalised density of the form
$p(x)=Ae^{Nf(x)}$, with a global maximum at $x_{0}$, then these averages
reduce to $g(x_{0})$. I.e., we can just evaluate the function to be averaged
at the ‘saddle point’, $x_{0}$.

We remind the reader again that we have here only considered real integrands,
and that the technique known as Laplace’s method, steepest descent, or saddle-
point integration extends much further. Information be found in [45, 46, 47,
48], and/or many textbooks or sets of lecture notes on asymptotic analysis and
methods.

##  4 The idea of generating functions and generating functionals

We now introduce the idea of generating functions (for static random
variables), and then generating functionals (for stochastic processes in
time).

###  4.1 Generating functions for random variables

####  4.1.1 Scalar random variables

Suppose $X$ is a real-valued random variable, with probability density
$p(\cdot)$. The generating function is then defined as

$Z(\psi)=\int dx~{}p(x)e^{ix\psi}.$ (47)

This is a function of the real variable $\psi$, and the function $Z(\cdot)$ is
(up to a pre-factor) nothing else than the Fourier transform of $p(\cdot)$.

The generating function has the following property

$Z(\psi=0)=1,$ (48)

this follows from the normalisation $\int dx~{}p(x)=1$.

We also have

$(-i)^{n}\left.\frac{d^{n}}{d\psi^{n}}Z(\psi)\right|_{\psi=0}=\left\langle{x^{n}}\right\rangle,$
(49)

where $\left\langle{f(x)}\right\rangle\equiv\int dx~{}p(x)f(x)$ denotes
averages of over $x$. This is why $Z(\cdot)$ is called the (moment) generating
function. If the function $Z(\psi)$ is known, the moments of the random
variable $X$ can be obtained from the derivatives of $Z(\psi)$ with respect to
$\psi$, evaluated at $\psi=0$. Conversely, $Z$ can be expressed as its Taylor
series,

$Z(\psi)=\sum_{n}\frac{(i\psi)^{n}}{n!}\left\langle{x^{n}}\right\rangle.$ (50)

This can be seen from writing the exponential in Eq. (47) as its series
expansion. Therefore, if all moments $\left\langle{x^{n}}\right\rangle$ are
known, this fully determines $Z(\cdot)$, and hence the probability density
$p(\cdot)$.

####  4.1.2 Vector-valued random variables

The definition of a generating function generalises to vector-valued random
variables, say ${\mathbf{X}}=(x_{1},\dots,x_{N})$. We then define

$Z({\mbox{\boldmath$\psi$}})=\int dx_{1}\cdots
dx_{N}~{}p(x_{1},\dots,x_{N})\exp\left(i\sum_{i=1}^{N}x_{i}\psi_{i}\right),$
(51)

where ${\mbox{\boldmath$\psi$}}=(\psi_{1},\dots,\psi_{N})$.

Again we have $Z({\mbox{\boldmath$\psi$}}={\mathbf{0}})=1$ from the
normalisation of the distribution $p(x_{1},\dots,x_{N})$. Moments can now be
obtained as follows

$\left\langle{x_{1}^{k_{1}}\cdots
x_{N}^{k_{N}}}\right\rangle=(-i)^{k_{1}+\dots+k_{N}}\left.\frac{\partial^{k_{1}+\dots+k_{N}}}{\partial\psi_{1}^{k_{1}}\cdots\partial\psi_{N}^{k_{N}}}\right|_{{\mbox{\boldmath$\psi$}}=0}Z({\mbox{\boldmath$\psi$}}).$
(52)

###  4.2 Generating functionals for stochastic processes

####  4.2.1 Discrete-time processes

Suppose now that we have a discrete-time process $x(0),x(1),x(2),\dots$, i.e.,
the $x(t)$, $t=0,1,2,\dots$ are a sequence of random numbers.

The details of the process are determined by the exact rules by which $x(t)$
evolves in time, and this results in an overall joint distribution for the
$x(t)$ ($t=0,1,\dots$).

The generating function for a scalar process in discrete time is very similar
to the one in Eq. (51) for a vector-valued random variable,

$Z(\underline{\psi})=\int
d\underline{x}~{}p(\underline{x})\exp\left(i\sum_{t=0}^{\infty}x(t)\psi(t)\right).$
(53)

We note that the source-field $\underline{\psi}$ is now a function (of the
discrete time variable $t$). The notation $d\underline{x}$ is to be understood
as

$d\underline{x}=\prod_{t=0,1,2,\dots}dx(t).$ (54)

Suppose now that the process $x(\cdot)$ is Markovian. This means that the
distribution of the random variable $x(t+1)$ is determined by the value
$x(t)$, and that no information about $x(t^{\prime})$ at earlier times
$t^{\prime}<t$ is required. We write this in the following form

$x(t+1)=g[x(t),\xi(t)].$ (55)

This is to be read as follows. Assume, we have run the stochastic process up
to time $t$, i.e., we have a particular realisation for
$x(0),x(1),\dots,x(t)$. The next point $x(t+1)$ of this realisation is then
obtained from the value of $x(t)$ and a random element $\xi(t)$. The nature of
the function $g$ determines the details of the process.

Example (unbiased random walk):
Let us consider the unbiased random walk on the set of (positive and negative)
integers. At each step the walker hops to the left with probability $1/2$, or
to the right, also with probability $1/2$. This can be represented by a binary
random variable $\xi(t)$, taking values $\xi(t)=\pm 1$, each with probability
$1/2$, and by choosing $g[x(t),\xi(t)]=x(t)+\xi(t)$. The $\xi(t)$ are not
correlated in time.

Alternatively, $x(t)$ could be continuous (but we keep time discrete for the
time being). For example, the walker could make steps with a size drawn from a
Gaussian distribution. I.e., we’d again have $g[x(t),\xi(t)]=x(t)+\xi(t)$, but
the $\xi(t)$ are now Gaussian random variables.

For discrete-time processes Markovian processes of the form
$x(t+1)=g[x(t),\xi(t)]$ the generating function turns into

$\displaystyle Z(\underline{\psi})$ $\displaystyle=$ $\displaystyle\int
d\underline{x}~{}d\underline{\xi}~{}p_{0}[x(t=0)]~{}p(\underline{\xi})$ (56)
$\displaystyle\times\left[\prod_{t=0}^{\infty}\delta[x(t+1)-g(x(t),\xi(t)]\right]\exp\left(i\sum_{t=0}^{\infty}x(t)\psi(t)\right).$

The integral is over (i) the potentially random initial condition $x(0)$, (ii)
all subsequent states at later times [$x(1),x(2),\dots$], and all realisations
of the noise process $\underline{\xi}$, i.e, over
$\xi(0),\xi(1),\xi(2),\dots$. The notation $d\underline{x}$ stands for
$dx(0)dx(1)dx(2)\dots$, and similarly for $d\underline{\xi}$.

The $\\{x(t),\xi(t)\\}$ are of course not independent from each other, instead
we need to impose $x(t+1)=g[x(t),\xi(t)]$. That’s what the product of delta
functions in the first term on the second line in Eq. (56) does. The last term
finally is the standard ‘source term’.

We can now use Eq. (21) to write

$\displaystyle Z(\underline{\psi})$ $\displaystyle=$ $\displaystyle\int
d\underline{\xi}{\cal D}\underline{x}~{}{\cal
D}\widehat{}\underline{x}~{}p_{0}[x(0)]p(\underline{\xi})$ (57)
$\displaystyle\times\exp\left(i\sum_{t=0}^{\infty}\widehat{x}(t)\left\\{x(t+1)-g[x(t),\xi(t)]\right\\}\right)$
$\displaystyle\times\exp\left(i\sum_{t=0}^{\infty}x(t)\psi(t)\right),$

where we have introduced the notation

${\cal D}\underline{x}=\prod_{t=0,1,2,\dots}\frac{dx(t)}{\sqrt{2\pi}},$ (58)

and similarly ${\cal
D}\widehat{}\underline{x}=\prod_{t=0,1,2,\dots}\frac{d\widehat{x}(t)}{\sqrt{2\pi}}$.

Depending on the properties of the noise and on the nature of the function
$g(\cdot,\cdot)$ this can be evaluated further. For example, let us consider
the Gaussian random walk

$g[x(t),\xi(t)]=x(t)+\xi(t),$ (59)

with independent Gaussian random variables $\xi(t)$, with mean $\mu$ and
variance $\sigma^{2}$. We can then write $\xi(t)=\mu+\sigma z(t)$, with $z(t)$
independent standard Gaussian random variables with mean zero and unit
variance.

Then

$\displaystyle Z(\underline{\psi})$ $\displaystyle=$ $\displaystyle\int{\cal
D}\underline{z}{\cal D}\underline{x}{\cal
D}\widehat{}\underline{x}~{}p_{0}[x(0)]\exp\left(-\frac{1}{2}\sum_{t=0}^{\infty}z(t)^{2}\right)\exp\left(i\sum_{t=0}^{\infty}\widehat{x}(t)\left\\{x(t+1)-x(t)-\mu-\sigma
z(t)\right\\}\right)$ (60)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\exp\left(i\sum_{t=0}^{\infty}x(t)\psi(t)\right),$

with ${\cal D}\underline{z}=\prod_{t=0}^{\infty}\frac{dz(t)}{\sqrt{2\pi}}$.

Given that the $z(t)$ are all independent from one another, we can now carry
out the average over each of the $z(t)$ separately. We have factors of the
form
$\int\frac{dz(t)}{\sqrt{2\pi}}\exp\left[-\frac{1}{2}z(t)^{2}-i\sigma\widehat{x}(t)z(t)\right]$.
These can be evaluated using the identity in Eq. (31), and we obtain

$\int\frac{dz(t)}{\sqrt{2\pi}}\exp\left(-\frac{1}{2}z(t)^{2}-i\sigma\widehat{x}(t)z(t)\right)=\exp\left(-\frac{1}{2}\sigma^{2}\widehat{x}(t)^{2}\right).$
(61)

The generating function in Eq. (60) then ‘simplifies’ to

$\displaystyle Z(\underline{\psi})$ $\displaystyle=$ $\displaystyle\int{\cal
D}\underline{x}{\cal D}\widehat{}\underline{x}~{}p_{0}[x(0)]$ (62)
$\displaystyle\times\exp\left(i\sum_{t=0}^{\infty}\widehat{x}(t)\left\\{x(t+1)-x(t)-\mu\right\\}-\frac{1}{2}\sigma^{2}\sum_{t=0}^{\infty}\widehat{x}(t)^{2}+i\sum_{t=0}^{\infty}x(t)\psi(t)\right).$

Now, this expression is admittedly not particularly enlightening, and not
immediately useful. But at least we have a chance to practice the basic
manipulation of generating functionals, and our skills in Gaussian
integration.

Exercise:
By successively performing the integral over the $\\{\widehat{x}(t)\\}$, and
then the $\\{x(t)\\}$, show that $Z(\underline{\psi}=\underline{0})=1$ for
$Z(\underline{\psi})$ in Eq. (62).

####  4.2.2 Continuous-time stochastic processes

Let us now look at continuous-time processes. More specifically, we assume
that $x(t)$ follows a stochastic differential equation (SDE) of the form

$\dot{x}=g(x)+\sigma\xi(t),$ (63)

where $\xi(t)$ is Gaussian white noise of mean zero, and unit variance, i.e.,

$\left\langle{\xi(t)}\right\rangle=0,~{}~{}~{}\left\langle{\xi(t)\xi(t^{\prime})}\right\rangle=\delta(t-t^{\prime}).$
(64)

In order to set up the generating functional, we will discretise time into
steps of size $\Delta t$. We then have

$x(t+\Delta)=x(t)+\Delta\times g[x(t)]+\sqrt{\Delta}\sigma z(t),$ (65)

where the $z(t)$ are independent standard Gaussian random variables (mean
zero, variance one). (If you are not familiar with the discretisation of
stochastic differential equations I recommend the book by Jacobs ‘Stochastic
Processes for Physicists’ [49]).

We can read off the generating functional for this discretised version from
Eq. (62). All we need to do is to replace $\sigma^{2}\to\Delta\sigma^{2}$, and
to adapt the term multiplying $\widehat{x}(t)$ in the curly bracket (this term
comes from the deterministic part of the process). We find

$\displaystyle Z(\underline{\psi})$ $\displaystyle=$ $\displaystyle\int{\cal
D}\underline{x}{\cal
D}\widehat{}\underline{x}~{}p_{0}[x(0)]\exp\left(i\sum_{t=0}^{\infty}\widehat{x}(t)\left\\{x(t+\Delta)-x(t)-\Delta
g[x(t)]\right\\}\right.$ (66) $\displaystyle\hskip
50.00008pt\left.-\frac{1}{2}\sigma^{2}\Delta\sum_{t=0}^{\infty}\widehat{x}(t)^{2}+i\sum_{t=0}^{\infty}x(t)\psi(t)\right),$

After minor re-arrangements we then have

$\displaystyle Z(\underline{\psi})$ $\displaystyle=$ $\displaystyle\int{\cal
D}\underline{x}{\cal
D}\widehat{}\underline{x}~{}p_{0}[x(0)]\exp\left(i\Delta\sum_{t=0}^{\infty}\widehat{x}(t)\left\\{\frac{x(t+\Delta)-x(t)}{\Delta}-g[x(t)]\right\\}\right.$
(67) $\displaystyle\hskip
50.00008pt\left.-\frac{1}{2}\sigma^{2}\Delta\sum_{t=0}^{\infty}\widehat{x}(t)^{2}+i\sum_{t=0}^{\infty}x(t)\psi(t)\right),$

As a next step, we re-name $\psi(t)\to\Delta\psi(t)$ (this is merely a
question of choosing the ‘scale’ for $\psi$). The expression for the
generating functional then becomes

$\displaystyle Z(\underline{\psi})$ $\displaystyle=$ $\displaystyle\int{\cal
D}\underline{x}{\cal
D}\widehat{}\underline{x}~{}p_{0}[x(0)]\exp\left(i\Delta\sum_{t=0}^{\infty}\widehat{x}(t)\left\\{\frac{x(t+\Delta)-x(t)}{\Delta}-g[x(t)]\right\\}\right.$
(68) $\displaystyle\hskip
50.00008pt\left.-\frac{1}{2}\sigma^{2}\Delta\sum_{t=0}^{\infty}\widehat{x}(t)^{2}+i\Delta\sum_{t=0}^{\infty}x(t)\psi(t)\right),$

We can now take the continuum limit $\Delta\to 0$ (we won’t worry too much
about what exactly happens to the measure ${\cal D}\underline{x}{\cal
D}\widehat{}\underline{x}$ in this limit). Taking $\Delta\to 0$ turns the sums
into integrals. We find

$\displaystyle Z(\underline{\psi})$ $\displaystyle=$ $\displaystyle\int{\cal
D}\underline{x}{\cal
D}\widehat{}\underline{x}~{}p_{0}[x(0)]\exp\left(i\int_{0}^{\infty}dt~{}\widehat{x}(t)\left\\{\dot{x}(t)-g[x(t)]\right\\}\right.$
(69) $\displaystyle\hskip
50.00008pt\left.-\frac{1}{2}\sigma^{2}\int_{0}^{\infty}dt~{}\widehat{x}(t)^{2}+i\int_{0}^{\infty}dt~{}x(t)\psi(t)\right).$

###  4.3 Correlation and response functions for discrete-time processes

####  4.3.1 Averages over realisations of the underlying stochastic process

In the following we will write $\left\langle{\cdots}\right\rangle$ for
averages over the stochastic process we are studying. In this short subsection
we’ll briefly discuss how this average relates to the generating functional
for the process.

Take for example the discrete-time process in Eq. (55),
$x(t+1)=x(t)+g[x(t),\xi(t)]$. The average of a function(al) $f(\underline{x})$
is then given by

$\displaystyle\left\langle{f(\underline{x})}\right\rangle$ $\displaystyle=$
$\displaystyle\int d\underline{\xi}{\cal D}\underline{x}{\cal
D}\widehat{}\underline{x}~{}p_{0}[x(0)]p(\underline{\xi})~{}f(\underline{x})$
(70)
$\displaystyle\times\exp\left(i\sum_{t=0}^{\infty}\widehat{x}(t)\left\\{x(t+1)-g[x(t),\xi(t)]\right\\}\right).$

The generating functional on the other hand is

$\displaystyle Z[\underline{\psi}]$ $\displaystyle=$ $\displaystyle\int
d\underline{\xi}{\cal D}\underline{x}{\cal
D}\widehat{}\underline{x}~{}p_{0}[x(0)]p(\underline{\xi})$ (71)
$\displaystyle\times\exp\left(i\sum_{t=0}^{\infty}\widehat{x}(t)\left\\{x(t+1)-g[x(t),\xi(t)]\right\\}\right)\exp\left(i\int
dt\psi(t)x(t)\right)$

Suppose now that $f$ is a simple function(al), for example
$f(\underline{x})=x(t_{1})x(t_{2})$ for two fixed times $t_{1}$ and $t_{2}$.
It is then clear that the average of $f$ can be generated from the generating
functional as derivatives with respect to $\psi(t_{1})$ and $\psi(t_{2})$, and
by subsequently setting $\underline{\psi}=0$. We’ll make this a little more
formal in the next subsections.

To do this, we extend the definition of an ‘average’ to include objects
involving $\widehat{}\underline{x}$, and write for example

$\displaystyle\left\langle{f(\underline{x},\widehat{}\underline{x})}\right\rangle$
$\displaystyle=$ $\displaystyle\int d\underline{\xi}{\cal D}\underline{x}{\cal
D}\widehat{}\underline{x}~{}p_{0}[x(0)]p(\underline{\xi})~{}f(\underline{x},\widehat{}\underline{x})$
(72) $\displaystyle\times\exp\left(i\int
dt\widehat{x}(t)\left\\{\dot{x}(t)-g(x(t))-\sigma\xi(t)\right\\}\right).$

####  4.3.2 Derivatives of the generating functional and correlation functions

With $\left\langle{\dots}\right\rangle$ the average over realisations of the
process, the generating function (or functional) is of the general form [see
e.g. Eqs. (47) and (53)]

$Z[\underline{\psi}]=\left\langle{e^{i\underline{x}\cdot\underline{\psi}}}\right\rangle,$
(73)

where $\underline{x}\cdot\underline{\psi}$ represents $\int dt\psi(t)x(t)$ in
the continuous case, and $\sum_{t}\psi(t)x(t)$ if time is discrete.

We first focus on the discrete-time case,
$Z(\underline{\psi})=\left\langle{e^{i\sum_{t}\psi(t)x(t)}}\right\rangle$. It
is then easy to see that moments of the $\\{x(t)\\}$ can be obtained by taking
derivatives with respect to the $\\{\psi(t)\\}$, evaluated at
$\underline{\psi}=0$. This is similar to what we have seen in Eq. (49) for
scalar and in Eq. (52) for vector-valued random variables.

For example, we have

$\left\langle{x(t)}\right\rangle=-i\left.\frac{\partial}{\partial\psi(t)}Z[\underline{\psi}]\right|_{\underline{\psi}=\underline{0}}$
(74)

To see this, use the identity

$\frac{\partial}{\partial\psi(t)}\sum_{t^{\prime}}\psi(t^{\prime})x(t^{\prime})=x(t).$
(75)

Similarly, we also have

$\left\langle{x(t)x(t^{\prime})}\right\rangle=-\left.\frac{\partial^{2}}{\partial\psi(t)\partial\psi(t^{\prime})}Z[\underline{\psi}]\right|_{\underline{\psi}=\underline{0}}$
(76)

This object is the correlation function of the process $x(\cdot)$.

####  4.3.3 Response functions

We now look at the discrete-time process in Eq. (55), again, but we add a
‘perturbation field’, $h(t)$, i.e., we start from

$x(t+1)=g[x(t),\xi(t)]+h(t).$ (77)

We would now like to know things such as ‘How does $x(t_{2})$ react to a
perturbation $h(t_{1})$ at an earlier time?’

The generating functional for the perturbed process can be obtained as a minor
modification of Eq. (53):

$\displaystyle Z(\underline{\psi},\underline{h})$ $\displaystyle=$
$\displaystyle\int d\underline{\xi}{\cal D}\underline{x}{\cal
D}\widehat{}\underline{x}~{}p_{0}[x(0)]p(\underline{\xi})\exp\left(i\sum_{t=0}^{\infty}\widehat{x}(t)\left\\{x(t+1)-g[x(t),\xi(t)]-h(t)\right\\}\right)$
(78) $\displaystyle\hskip
50.00008pt\times\exp\left(i\sum_{t=0}^{\infty}x(t)\psi(t)\right),$

Response functions are of the type

$G(t,t^{\prime})=\frac{\partial}{\partial
h(t^{\prime})}\left\langle{x(t)}\right\rangle.$ (79)

We recall that $\left\langle{\cdots}\right\rangle$ is an average over
realisations of the stochastic process. The quantity $G(t,t^{\prime})$
therefore describes how the field $x(t)$ ‘typically’ reacts to a perturbation
at time $t^{\prime}$. More precisely, the matrix $\underline{\underline{G}}$
captures linear response to small perturbations. By causality we must have
$G(t,t^{\prime})=0$ for $t^{\prime}>t$.

Using

$\displaystyle\left\langle{x(t)}\right\rangle$ $\displaystyle=$
$\displaystyle\int d\underline{\xi}{\cal D}\underline{x}{\cal
D}\widehat{}\underline{x}~{}p_{0}[x(0)]p(\underline{\xi})~{}x(t)$ (80)
$\displaystyle\times\exp\left(i\sum_{t^{\prime\prime}=0}^{\infty}\widehat{x}(t^{\prime\prime})\left\\{x(t^{\prime\prime}+1)-g[x(t^{\prime\prime}),\xi(t^{\prime\prime})]-h(t^{\prime\prime})\right\\}\right)$

and Eq. (79) one sees that

$G(t,t^{\prime})=-i\left\langle{x(t)\widehat{x}(t^{\prime})}\right\rangle.$
(81)

####  4.3.4 Moments of conjugate variables vanish

We’ll now derive an important fact, namely, averages of objects involving only
conjugate fields vanish. We will use this at various points later on.

We start by noting that

$Z(\underline{\psi}=\underline{0},\underline{h})=1$ (82)

for any fixed choice of the perturbation field $\underline{h}$. This can
either be seen by carrying out the $\widehat{x}$-integration in Eq. (78)
(restoring the original $\delta$-functions), or perhaps easier directly from
the definition
$Z(\underline{\psi})=\left\langle{e^{i\underline{\psi}\cdot\underline{x}}}\right\rangle$.
This definition is valid for any stochasic process, so also the one in Eq.
(77), including the field $\underline{h}$. (For any fixed choice of the
function $h(t)$, we could absorb the perturbation field into $g$.) It then
follows that
$Z(\underline{\psi}=\underline{0},h)=\left\langle{1|\underline{h}}\right\rangle$,
where the average $\left\langle{\cdots|\underline{h}}\right\rangle$ is over
realisations of the stochastic process in Eq. (77) with the (fixed)
perturbation field $h(t)$ included. Thus,
$Z(\underline{\psi}=\underline{0},h)$ is the average of one, and therefore
equal to one.

Eq. (82) in turn implies

$\frac{\partial}{\partial
h(t)}Z(\underline{\psi}=\underline{0},\underline{h})=0.$ (83)

Using Eq. (78), the derivative on the right-hand side is recognised as
$-i\left\langle{\widehat{x}(t)|\underline{h}}\right\rangle$, and we therefore
conclude that $\left\langle{\widehat{x}(t)|\underline{h}}\right\rangle=0$.
This is true for all choices of the perturbation field $h(\cdot)$, so in
particular also if $h(t)\equiv 0$. We conclude

$\left\langle{\widehat{x}(t)}\right\rangle=0,~{}~{}\forall t.$ (84)

Higher-order derivatives of $Z(\underline{\psi}=\underline{0},\underline{h})$
with respect to the components of $\underline{h}$ are also zero. Following
similar steps as above, one then sees that any moment involving only conjugate
variables $\widehat{x}(\cdot)$ vanishes, e.g.

$\left\langle{\widehat{x}(t)\widehat{x}(t^{\prime})}\right\rangle=0.$ (85)

Given that any (sensible) function $f$ of the $\\{\widehat{x}(t)\\}$ can be
expressed in a Taylor series, we can therefore conclude that

$\left\langle{f(\\{\widehat{x}(t)\\}}\right\rangle=0$ (86)

for all such functions.

###  4.4 Continuous-time processes and functional derivatives

The conventional partial derivative of a function of the form
$f(a_{1}x_{1}+\dots+a_{N}x_{N})$ is as follows,

$\frac{\partial}{\partial
x_{i}}f(a_{1}x_{1}+\dots+a_{N}x_{N})=f^{\prime}(a_{1}x_{1}+\dots+a_{N}x_{N})a_{i},$
(87)

where $f^{\prime}(y)=\frac{df(y)}{dy}$.

Imagine now that the index $i$ in $\sum_{i}a_{i}x_{i}$ is continuous (we’ll
call it $t$). Then we’d be dealing with an object of the form $f\left(\int
dta(t)x(t)\right)$. The functional derivative then works just as in the
discrete case, it picks out the right ‘term’ in the integral. We write this as

$\frac{\delta}{\delta x(t)}f\left(\int
dt^{\prime}a(t^{\prime})x(t^{\prime})\right)=f^{\prime}\left(\int
dt^{\prime}a(t^{\prime})x(t^{\prime})\right)a(t).$ (88)

For example

$\frac{\delta}{\delta\psi(t)}\int
D\underline{x}~{}e^{g(\underline{x})}~{}e^{i\int
dt^{\prime}\psi(t^{\prime})x(t^{\prime})}=i\int
D\underline{x}~{}e^{g(\underline{x})}~{}e^{i\int
dt^{\prime}\psi(t^{\prime})x(t^{\prime})}x(t),$ (89)

and

$\frac{\delta}{\delta\psi(t)}\ln\int
D\underline{x}~{}e^{g(\underline{x})}~{}e^{i\int
dt^{\prime}\psi(t^{\prime})x(t^{\prime})}=\frac{i\int
D\underline{x}~{}e^{g(\underline{x})}~{}e^{i\int
dt^{\prime}\psi(t^{\prime})x(t^{\prime})}x(t)}{\int
D\underline{x}~{}e^{g(\underline{x})}~{}e^{i\int
dt^{\prime}\psi(t^{\prime})x(t^{\prime})}}.$ (90)

With this definition, the steps in Sec. 4.3 can be repeated with modest
modifications (amounting to replacing sums over $t$ with integrals, and the
symbol $\partial$ with the symbol $\delta$). We arrive at

$\left\langle{x(t)x(t^{\prime})}\right\rangle=-\left.\frac{\delta^{2}}{\delta\psi(t)\delta\psi(t^{\prime})}Z[\underline{\psi}]\right|_{\underline{\psi}=\underline{0}}$
(91)

as well as

$G(t,t^{\prime})=\frac{\delta}{\delta
h(t^{\prime})}\left\langle{x(t)}\right\rangle=-i\left\langle{x(t)\widehat{x}(t^{\prime})}\right\rangle.$
(92)

As in discrete time, we find that averages of objects involving only the
conjugate field $\\{\widehat{x}(t)\\}$ vanish.

##  5 Toy example of a generating-functional calculation: Interacting
processes without disorder

We’ll now use generating functionals to derive a dynamic mean-field
description for a problem involving interacting stochastic processes. In this
initial example, there is no disorder. This is much simpler, but nevertheless
instructive. The full calculation for the disordered Lotka–Volterra problem
then follows in Secs. 6-8.

The structure of this section is as follows (see also the illustration in Fig.
1). We set up the problem in Sec. 5.1. A heuristic derivation of the dynamic
mean field theory is then presented in Sec. 5.2. Secs. 5.3-5.5 contain the
generating-functional analysis of the model. More precisely, the generating
functional is set up in Sec. 5.3. We present a first derivation of the dynamic
mean-field process from the generating functional in Sec. 5.4. One step of the
derivation is somewhat sloppy, but nonetheless the overall direction is right,
and it is instructive to look at this. A proper derivation is then presented
in Sec. 5.5. We note that all three approaches (heuristic derivation, sloppy
analysis of the generating functional saddle point, and the proper saddle-
point analysis) lead to the same dynamic mean field process.

Figure 1: Illustration of the structure of this section. We first introduce
the model (5.1). The rest of the section then describes three routes all
leading to the same dynamic mean field process. The first approach is
heuristic and does not use generating functionals (Sec. 5.2). Alternatively
one carries out a generating-functional analysis, leading to a saddle-point
problem. In Sec. 5.4 we first describe a simpler, but defective method of
extracting the effective dynamics from this saddle point. A more rigorous
derivation can then be found in Sec. 5.5.

###  5.1 Setup

We choose the dynamics

$\dot{x}_{i}=ax_{i}+g\left(\frac{1}{N}\sum_{j}x_{j}\right)+\xi_{i}(t),$ (93)

where $i=1,\dots,N$, and where $a$ is a constant, and $g(\cdot)$ can be any
(real-valued) function. We have included Gaussian white noise variables,
$\\{\xi_{i}(t)\\}$, with zero mean and variance $\sigma^{2}$, and no
correlations between components, i.e.

$\left\langle{\xi_{i}(t)\xi_{j}(t^{\prime})}\right\rangle=\sigma^{2}\delta_{ij}\delta(t-t^{\prime}).$
(94)

There is no requirement for the $\\{x_{i}(t)\\}$ to be positive.

###  5.2 Heuristic derivation of the evolution equation for first moment in
the thermodynamic limit

We first present a heuristic derivation of a closed equation for the relevant
macroscopic dynamic order parameter. This is similar to the ad-hoc derivation
of the mean-field approach in Sec. 2.

We write $M(t)=\frac{1}{N}\sum_{j}x_{j}(t)$. From Eq. (93) we then find

$\dot{M}=aM(t)+g[M(t)]+\frac{1}{N}\sum_{i}\xi_{i}(t)$ (95)

The quantity $M(t)$ is a random variable, because of two elements of
randomness: One is the noise in Eq. (93), i.e., the $\\{\xi_{i}(t)\\}$. The
other comes (potentially) from the initial conditions for $\mathbf{x}(t=0)$
which can also be random.

In the thermodynamic limit, $N\to\infty$, the noise in Eq. (95) averages out,
we have $\frac{1}{N}\sum_{i}\xi_{i}(t)\to 0$ via the Central Limit Theorem. We
then have the following deterministic evolution for $M(t)$,

$\dot{M}=aM(t)+g[M(t)],$ (96)

with a potentially random initial condition for the $\\{x_{i}(t=0)\\}$. If the
$\\{x_{i}(t=0)\\}$ are independent and identically distributed,
$p[\mathbf{x}(0)]=\prod_{i=1}^{N}p_{0}[x_{i}(0)]$, we find

$\displaystyle M(0)$ $\displaystyle=$
$\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{i}x_{i}(0)=\int
dx~{}p_{0}(x)x,$ (97)

via the law of large numbers, assuming the first moment of the distribution
$p_{0}$ exists. Eqs. (96) and (97) fully describe the time-evolution of the
macroscopic order parameter $M(t)$.

###  5.3 Generating-functional analysis I: Setup and saddle-point problem

We now re-derive this via generating functionals. Of course, this is an
overkill for this simple example, but it is useful to see the generating-
functional method in action.

####  5.3.1 Setting up the generating functional

The dynamical generating functional is defined as

$Z[\underline{{\mbox{\boldmath$\psi$}}}]=\int
D\underline{\bm{\xi}}D\underline{\mathbf{x}}~{}p(\underline{\bm{\xi}})\,p_{0}[\mathbf{x}(0)]\,\delta(\mbox{equations
of motion})e^{i\sum_{i}\int dt~{}x_{i}(t)\psi_{i}(t)},$ (98)

with $p(\underline{\bm{\xi}})$ the distribution of the noise variables, and
$p_{0}[\mathbf{x}(0)]$ the distribution of the initial conditions for the
$\\{x_{i}\\}$ at time $t=0$. We have introduced the notation

$D\underline{\mathbf{x}}=\prod_{i=1}^{N}\prod_{t=0,1,2...}dx_{i}(t),$ (99)

and similarly for $D\underline{\bm{\xi}}$. When we say ‘equations of motion’,
we mean the dynamics in Eq. (93), so

$\delta(\mbox{equations of
motion})=\prod_{it}\delta\left[\dot{x}_{i}-ax_{i}+g\left(\frac{1}{N}\sum_{j}x_{j}\right)+\xi_{i}(t)\right].$
(100)

We now proceed with various manipulations in the generating functional.
Expressing the delta-functions as Fourier transforms, we have

$\displaystyle Z[\underline{{\mbox{\boldmath$\psi$}}}]$ $\displaystyle=$
$\displaystyle\int D\underline{\bm{\xi}}{\cal D}\underline{\mathbf{x}}{\cal
D}\widehat{}\underline{\mathbf{x}}\,p(\underline{\bm{\xi}})\,p_{0}[\mathbf{x}(0)]\,\exp\left(i\sum_{i}\int
dt~{}x_{i}(t)\psi_{i}(t)\right)$ (101)
$\displaystyle\times\exp\Bigg{\\{}i\sum_{i}\int
dt\Bigg{[}\widehat{x}_{i}(t)\Big{(}\dot{x}_{i}(t)-ax_{i}(t)-g\left(\frac{1}{N}\sum_{j}x_{j}(t)\right)-\xi_{i}(t)\Big{)}\Bigg{]}\Bigg{\\}},$

where

${\cal
D}\underline{\mathbf{x}}=\prod_{i=1}^{N}\prod_{t=0,1,2...}\frac{dx_{i}(t)}{\sqrt{2\pi}}.$
(102)

####  5.3.2 Introduction of macroscopic order parameter

Next, for any fixed time $t$, we can insert unity in the following form

$1=\int_{-\infty}^{\infty}\frac{dM(t)d\widehat{M}(t)}{2\pi/N}~{}e^{iN\widehat{M}(t)\left[M(t)-N^{-1}\sum_{i}x_{i}(t)\right]}$
(103)

This may look a little weird at first. The purpose of this is to introduce the
macroscopic order parameter $M(t)=\frac{1}{N}\sum_{i}x_{i}(t)$ into the
calculation. It is important to realise that carrying out the integration over
$\widehat{M}(t)$ reduces Eq. (103) to $1=\int
dM(t)\,\delta\left(M(t)-N^{-1}\sum_{i}x_{i}(t)\right)$ [see also Eq. (21)].

Inserting Eq. (103) into the generating functional for all times $t$, we get

$\displaystyle Z[{\mbox{\boldmath$\psi$}}]$ $\displaystyle=$
$\displaystyle\int D\underline{\bm{\xi}}\,{\cal D}\underline{\mathbf{x}}{\cal
D}\widehat{}\underline{\mathbf{x}}\,{\cal
D}[\underline{M},\widehat{}\underline{M}]\,p(\underline{\bm{\xi}})\,p_{0}[\mathbf{x}(0)]\,\exp\left(i\int
dt\,\widehat{M}(t)\left[NM(t)-\sum_{i}x_{i}(t)\right]\right)$ (104)
$\displaystyle\exp\Bigg{\\{}i\sum_{i}\int
dt\Bigg{[}\widehat{x}_{i}(t)\Big{(}\dot{x}_{i}(t)-ax_{i}(t)-g\left(\frac{1}{N}\sum_{j}x_{j}(t)\right)-\xi_{i}(t)\Big{)}\Bigg{]}\Bigg{\\}}$
$\displaystyle\times\exp\left(i\sum_{i}\int dt~{}x_{i}(t)\psi_{i}(t)\right),$

with the notation

${\cal
D}[\underline{M},\widehat{}\underline{M}]=\prod_{t=0,1,2,\dots}\frac{dM(t)d\widehat{M}(t)}{2\pi/N}.$
(105)

Given that the $\widehat{}\underline{M}$-integration leads to delta functions,
$\delta\left[NM(t)-\sum_{i}x_{i}(t)\right]$, for all times $t$, we can replace
$N^{-1}\sum_{j}x_{j}(t)$ with $M(t)$ in the second exponential. This gives

$\displaystyle Z[{\mbox{\boldmath$\psi$}}]$ $\displaystyle=$
$\displaystyle\int D\underline{\bm{\xi}}\,{\cal
D}\underline{\mathbf{x}}\,{\cal D}\widehat{}\underline{\mathbf{x}}\,{\cal
D}[\underline{M},\widehat{}\underline{M}]\,p(\underline{\bm{\xi}})\,p_{0}[\mathbf{x}(0)]\,\exp\left(iN\int
dt\,\widehat{M}(t)\left[M(t)-N^{-1}\sum_{i}x_{i}(t)\right]\right)$
$\displaystyle\times\exp\Bigg{\\{}i\sum_{i}\int
dt\Bigg{[}\widehat{x}_{i}(t)\Big{(}\dot{x}_{i}(t)-ax_{i}(t)-g[M(t)]-\xi_{i}(t)\Big{)}\Bigg{]}\Bigg{\\}}$
$\displaystyle\times\exp\left(i\sum_{i}\int dt~{}x_{i}(t)\psi_{i}(t)\right).$

From now on, we assume that the random variables $x_{i}(0)$ ($i=1,\dots,N$)
are independent and identically distributed (iid). The distribution
$p_{0}[\mathbf{x}(0)]$ then factorises, and in slight abuse of notation
(double use of $p_{0}$), we write it in the form
$p_{0}[\mathbf{x}(0)]=\prod_{i}p_{0}[x_{i}(0)]$.

####  5.3.3 Formulation as a saddle-point problem

We now re-organise the terms in the generating functional in Eq.
(LABEL:eq:gf004). More precisely, we separate the contributions containing
only macroscopic objects [i.e., $M(t)$ and $\widehat{M}(t)$] from the terms
involving individual microscopic variables, $x_{i}(t)$ and
$\widehat{x}_{i}(t)$.

We write the generating functional in the following form

$\displaystyle Z[{\mbox{\boldmath$\underline{\psi}$}}]$ $\displaystyle=$
$\displaystyle\int{\cal
D}[\underline{M},\widehat{}\underline{M}]\exp\left(N\left(\Psi[\underline{M},\widehat{}\underline{M}]+\Omega[\underline{M},\widehat{}\underline{M},\underline{{\mbox{\boldmath$\psi$}}}]\right)\right),$
(107)

with

$\Psi[\underline{M},\widehat{}\underline{M}]=i\int dt\,\widehat{M}(t)M(t),$
(108)

and

$\displaystyle\Omega[\underline{M},\widehat{}\underline{M},{\mbox{\boldmath$\underline{\psi}$}}]$
$\displaystyle=$ $\displaystyle\frac{1}{N}\sum_{i}\ln\Bigg{\\{}\int{\cal
D}\underline{\xi}_{i}{\cal D}\underline{x}_{i}{\cal
D}\widehat{}\underline{x}_{i}~{}p(\underline{\xi}_{i})\,p_{0}[x_{i}(0)]$ (109)
$\displaystyle\times\exp\left[i\int
dt\left[\widehat{x}_{i}(t)\left(\dot{x}_{i}(t)-ax_{i}(t)-g[M(t)]-\xi_{i}(t)\right)\right]\right]$
$\displaystyle\times\exp\left[-i\int
dt\widehat{M}(t)x_{i}(t)\right]\exp\left[i\int
dt~{}x_{i}(t)\psi_{i}(t)\right]\Bigg{\\}},$

where we have written ${\cal
D}\underline{x}_{i}=\prod_{t=0,1,2,\dots}\frac{dx_{i}(t)}{\sqrt{2\pi}}$.

The variables $\xi_{i}(t)$ are dummy variables, to be integrated over, so we
can drop the index $i$. We can then write

$\displaystyle\Omega[\underline{M},\widehat{}\underline{M},{\mbox{\boldmath$\underline{\psi}$}}]$
$\displaystyle=$ $\displaystyle\frac{1}{N}\sum_{i}\ln\Bigg{\\{}\int{\cal
D}\underline{\xi}_{i}{\cal D}\underline{x}_{i}{\cal
D}\widehat{}\underline{x}_{i}~{}p(\underline{\xi})\,p_{0}[x_{i}(0)]$ (110)
$\displaystyle\times\exp\left[i\int
dt\left[\widehat{x}_{i}(t)\left(\dot{x}_{i}(t)-ax_{i}(t)-g[M(t)]-\xi(t)\right)\right]\right]$
$\displaystyle\times\exp\left[-i\int
dt\widehat{M}(t)x_{i}(t)\right]\exp\left[i\int
dt~{}x_{i}(t)\psi_{i}(t)\right]\Bigg{\\}}.$

###  5.4 Generating functional analysis II: ‘Vanilla’ derivation of dynamic
mean field process

We will now proceed with the evaluation of the saddle point problem in Eq.
(107). For pedagogical reasons I will first present a simple, but slightly
incorrect version of this. I call this the ‘vanilla’ version of the
calculation (if you prefer you can call it ‘light’ or ‘lite’). The ideas used
in this simplified calculation are broadly right, and so is the final result,
but some steps along the way are not entirely justified. Nonetheless, I think
it is useful to first present this simplified procedure (which, as we will
see, is quite involved already). This is so that the a beginner in this area
can develop a general understanding, and see where all this is going. I’ll
then say what wasn’t so precise (Sec. 5.4.3), and in Sec. 5.5, I’ll present a
refined version of the calculation.

####  5.4.1 Saddle-point equations for the order parameters

OK, here now the ‘light’ version of the saddle-point calculation. The first
simplification we make is to restrict the discussion to source fields
$\psi_{i}(t)=\psi(t)$ for all $i$. There is nothing fishy about this, it just
restricts the values of $\underline{\psi}$ for which we evaluate the
generating functional.

Making this simplification makes all terms in the sum in Eq. (110) equal, and
we can drop the index $i$ in $x_{i}$ and $\widehat{x}_{i}$. The expression for
$\Omega$ reduces to

$\displaystyle\Omega[\underline{M},\widehat{}\underline{M},{\mbox{\boldmath$\underline{\psi}$}}]$
$\displaystyle=$ $\displaystyle\ln\int D\underline{\xi}\,{\cal
D}\underline{x}\,{\cal
D}\widehat{}\underline{x}~{}p(\underline{\xi})\,p_{0}[x(0)]$ (111)
$\displaystyle\times\exp\left[i\int
dt\left[\widehat{x}(t)\left(\dot{x}(t)-ax(t)-g[M(t)]-\xi(t)\right)\right]\right]$
$\displaystyle\times\exp\left[-i\int
dt\widehat{M}(t)x(t)\right]\exp\left[i\int dt~{}x(t)\psi(t)\right].$

We want to compute the integral in Eq. (107). We do this using Laplace’s
method (Sec. 3.5).

We extremise the quantity $\Psi+\Omega$ in terms of $\underline{M}$ and
$\widehat{}\underline{M}$, i.e., we set

$\displaystyle\frac{\delta}{\delta
M(t)}\left(\Psi[\underline{M},\widehat{}\underline{M}]+\Omega[\underline{M},\widehat{}\underline{M},{\mbox{\boldmath$\underline{\psi}$}}]\right)$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\frac{\delta}{\delta\widehat{M}(t)}\left(\Psi[\underline{M},\widehat{}\underline{M}]+\Omega[\underline{M},\widehat{}\underline{M},{\mbox{\boldmath$\underline{\psi}$}}]\right)$
$\displaystyle=$ $\displaystyle 0,$ (112)

for all $t$.

From these two conditions and the definitions of $\Psi$ and $\Omega$ in Eqs.
(122) and (109) we get

$\displaystyle i\widehat{M}(t)$ $\displaystyle=$ $\displaystyle
ig^{\prime}[M(t)]\left\langle{\widehat{x}(t)}\right\rangle_{\Omega},$
$\displaystyle M(t)$ $\displaystyle=$
$\displaystyle\left\langle{x(t)}\right\rangle_{\Omega},$ (113)

where $g^{\prime}(y)=dg(y)/dy$, and where the ‘average’
$\left\langle{\cdots}\right\rangle_{\Omega}$ of any functional
$f(\underline{x},\widehat{}\underline{x})$ is defined as

$\displaystyle\left\langle{f(\underline{x},\widehat{}\underline{x})}\right\rangle_{\Omega}$
$\displaystyle=$ $\displaystyle\frac{1}{Z_{1}(\underline{\psi})}\int
D\underline{\xi}{\cal D}\underline{x}{\cal
D}\widehat{}\underline{x}~{}p_{0}[x(0]\,p(\underline{\xi})~{}f(\underline{x},\widehat{}\underline{x})$
(114) $\displaystyle\times\exp\left[i\int
dt\left[\widehat{x}(t)\left(\dot{x}(t)-ax(t)-g[M(t)]-\xi(t)\right)\right]\right]$
$\displaystyle\times\exp\left[-i\int
dt\widehat{M}(t)x(t)\right]\exp\left[i\int dt~{}x(t)\psi(t)\right],$

with

$\displaystyle Z_{1}(\underline{\psi})$ $\displaystyle=$ $\displaystyle\int
D\underline{\xi}{\cal D}\underline{x}{\cal
D}\widehat{}\underline{x}~{}p_{0}[x(0]\,p(\underline{\xi})~{}\exp\left[i\int
dt\left[\widehat{x}(t)\left(\dot{x}(t)-ax(t)-g[M(t)]-\xi(t)\right)\right]\right]$
(115) $\displaystyle\times\exp\left[-i\int
dt\widehat{M}(t)x(t)\right]\exp\left[i\int dt~{}x(t)\psi(t)\right].$

####  5.4.2 Elimination of hatted order parameter leads to single effective
particle process

Next we use the result of Sec. 4.3.4, namely that averages of hatted variables
are zero. This means that
$\left\langle{\widehat{x}(t)}\right\rangle_{\Omega}=0$. Using the first
relation in Eq. (5.4.1) we then find that $\widehat{M}(t)=0$.

This leads to

$\displaystyle\left.Z_{1}(\underline{\psi})\right|_{\widehat{}\underline{M}=0}$
$\displaystyle=$ $\displaystyle\int D\underline{\xi}{\cal D}\underline{x}{\cal
D}\widehat{}\underline{x}~{}p_{0}[x(0)]~{}p(\underline{\xi})$ (116)
$\displaystyle\times\exp\left[i\int
dt\left[\widehat{x}(t)\left(\dot{x}(t)-ax(t)-g[M(t)]-\xi(t)\right)\right]\right]$
$\displaystyle\times\exp\left[i\int dt~{}x(t)\psi(t)\right].$

This is recognised as the generating functional of the dynamics

$\dot{x}(t)=ax(t)+g[M(t)]+\xi(t).$ (117)

We will refer to this as an effective single-particle process. This is because
the process describes the ‘effective dynamics’ a typical particle ‘sees’ in
this system. It is a single-particle process, as there is no site index $i$
any longer.

From the second relation in Eq. (5.4.1) we have the self-consistency relation
$M(t)=\left\langle{x(t)}\right\rangle_{\Omega}$ . Using
$Z_{1}[\underline{\psi}=\underline{0}]=1$ (for $\widehat{M}(t)\equiv 0$) we
have

$\displaystyle\left\langle{f(\underline{x},\widehat{}\underline{x})}\right\rangle_{\Omega}\bigg{|}_{\widehat{}\underline{M}=\underline{0},~{}\underline{\psi}=\underline{0}}$
$\displaystyle=$ $\displaystyle\int D\underline{\xi}{\cal D}\underline{x}{\cal
D}\widehat{}\underline{x}~{}p_{0}[x(0]\,p(\underline{\xi})~{}f(\underline{x},\widehat{}\underline{x})$
(118) $\displaystyle\times\exp\left[i\int
dt\left[\widehat{x}(t)\left(\dot{x}(t)-ax(t)-g[M(t)]-\xi(t)\right)\right]\right].$

This describes averages over realisations of the process in Eq. (117).

Averaging both sides Eq. (117) over realisations (of $\xi$ and of the initial
value $x(0)$), i.e., carrying out the operation
$\left\langle{\cdots}\right\rangle_{\Omega}\big{|}_{\widehat{}\underline{M}=\underline{0},~{}\underline{\psi}=\underline{0}}$,
we find

$\dot{M}=aM+g(M),$ (119)

i.e., we recover the result of our ad-hoc calculation in Eq. (96).

####  5.4.3 The problems with this simplified calculation

Now, why do I call this the ‘vanilla’ version, or saddle point ‘light’? There
are several issues which we have swept under the carpet.

Elimination of $\widehat{M}(t)$ not properly justified:
The problem is that the justification for the step from Eq. (115) to (116)
above is not really right. Don’t get me wrong, it is true that
$\widehat{M}(t)=0$ at the saddle point (as we will see in Sec. 5.5), and the
result in Eq. (119) is also right. But the way we justified this is not
complete.

We simply quoted Sec. 4.3.4 (‘averages of hatted quantities are zero’) to
assert that $\left\langle{\widehat{x}(t)}\right\rangle_{\Omega}=0$, and then
used Eq. (5.4.1) to conclude that $\widehat{M}(t)=0$.

The problem is that an average of a conjugate variable (hatted quantity) only
vanishes if the average is over realisations of a bona fide stochastic
process. But this is not clear in Eq. (114). The object
$\left\langle{\cdots}\right\rangle_{\Omega}$ in Eq. (114) would be an average
over an actual stochastic dynamics if the term $\exp\left[-i\int
dt\widehat{M}(t)x(t)\right]$ wasn’t there, i.e., if all $\widehat{M}(t)$ were
zero (additionally we would have to set $\psi(t)\equiv 0$). If this was the
case then indeed $\left\langle{\cdots}\right\rangle_{\Omega}$ would be an
average over the process in Eq. (117), and therefore
$\left\langle{\widehat{x}(t)}\right\rangle_{\Omega}$ would be zero. But we
used $\left\langle{\widehat{x}(t)}\right\rangle_{\Omega}=0$ to argue that
$\widehat{M}(t)=0$, and we need $\widehat{M}(t)$ to vanish to show that
$\left\langle{\widehat{x}(t)}\right\rangle_{\Omega}=0$. So our argument is
circular.

This doesn’t mean that everything is wrong though. When we make the step from
Eq. (115) to (116) we could have simply made an ansatz and postulated that
$\widehat{M}(t)\equiv 0$ for all $t$. We could then have justified this a
posteriori, at the end of the calculation.

Generating functional $Z_{1}[\psi]$ for single effective particle not valid
for non-zero $\psi(t)$
This is related to the previous issue. In order to identify the operation
$\left\langle{\cdots}\right\rangle_{\Omega}$ as an average over a stochastic
process we need the source field to vanish ($\underline{\psi}(t)\equiv 0$).
But we then used the expression in Eq. (116) to argue that this is the
generating functional of the right single effective particle dynamics. In
order to make this final step, $\underline{\psi}$ has to be allowed to take
general values, resulting in another gap in our ‘vanilla’ derivation.

This is not a serious issue either. The measure in Eq. (118) does not contain
the source field, and it describes the right effective process.

Physical meaning of $M(t)$ not fully clear:
There is a further subtlety. We introduced the object
$M(t)=\frac{1}{N}\sum_{i}x_{i}(t)$ in the course of our calculation, but this
was done by inserting unity written in some complicated way [Eq. (103)]. At
that point $M(t)$ is mostly a mathematical short-hand that we used in order
not have to write $\frac{1}{N}\sum_{i}x_{i}(t)$ all the time. It is also
important to note that $M(t)$, in that form, is a stochastic quantity, and
only becomes non-random at the saddle point in the limit $N\to\infty$. It is
natural to assume that the saddle-point value of $M(t)$ is the same as
$\lim_{N\to\infty}\frac{1}{N}\sum_{i}\left\langle{x_{i}(t)}\right\rangle$ in
the original problem, where $\left\langle{\cdots}\right\rangle$ (without the
subscript $\Omega$) is an average over realisations of the process in Eq.
(93). This is indeed the case, but so far we have not formally proved this.

We address these points in the next section.

###  5.5 Generating functional analysis III: Proper derivation of dynamic
mean-field process

####  5.5.1 Generating functional for the problem with additional perturbation
fields

We return to the original problem [Eq. (93)], and add perturbation fields
$h_{i}(t)$,

$\dot{x}_{i}(t)=ax_{i}(t)+g\left(\frac{1}{N}\sum_{j}x_{j}(t)\right)+\xi_{i}(t)+h_{i}(t).$
(120)

These fields are not part of the model, and we will set them to zero
eventually. But we’ll need them along the way as a mathematical tool (to
generate moments of the conjugate field).

The generating functional for the system with perturbation fields is

$\displaystyle Z[{\mbox{\boldmath$\underline{\psi}$}}]$ $\displaystyle=$
$\displaystyle\int{\cal
D}[\underline{M},\widehat{}\underline{M}]\exp\left(N\left(\Psi[\underline{M},\widehat{}\underline{M}]+\Omega[\underline{M},\widehat{}\underline{M},\underline{{\mbox{\boldmath$\psi$}}}]\right)\right),$
(121)

with

$\Psi[\underline{M},\widehat{}\underline{M}]=i\int dt\,\widehat{M}(t)M(t),$
(122)

and

$\displaystyle\Omega[\underline{M},\widehat{}\underline{M},{\mbox{\boldmath$\underline{\psi}$}}]$
$\displaystyle=$ $\displaystyle\frac{1}{N}\sum_{i}\ln\Bigg{\\{}\int
D\underline{\xi}_{i}{\cal D}\underline{x}_{i}{\cal
D}\widehat{}\underline{x}_{i}~{}p(\underline{\xi})\,p_{0}[x_{i}(0)]$ (123)
$\displaystyle\times\exp\left[i\int
dt\left[\widehat{x}_{i}(t)\left(\dot{x}_{i}(t)-ax_{i}(t)-g[M(t)]-\xi_{i}(t)-h_{i}(t)\right)\right]\right]$
$\displaystyle\times\exp\left[-i\int
dt\widehat{M}(t)x_{i}(t)\right]\exp\left[i\int
dt~{}x_{i}(t)\psi_{i}(t)\right]\Bigg{\\}}.$

At the saddle point we again obtain

$\displaystyle i\widehat{M}(t)$ $\displaystyle=$ $\displaystyle
ig^{\prime}[M(t)]\left\langle{\widehat{x}(t)}\right\rangle_{\Omega},$
$\displaystyle M(t)$ $\displaystyle=$
$\displaystyle\left\langle{x(t)}\right\rangle_{\Omega},$ (124)

but the average $\left\langle{\cdots}\right\rangle_{\Omega}$ is now defined as

$\displaystyle\left\langle{f(\underline{x},\widehat{}\underline{x})}\right\rangle_{\Omega}$
$\displaystyle=$
$\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{i}\frac{1}{Z_{1}(\underline{\psi}_{i})}\int
D\underline{\xi}{\cal D}\underline{x}{\cal
D}\widehat{}\underline{x}~{}p_{0}[x(0]\,p(\underline{\xi})~{}f(\underline{x},\widehat{}\underline{x})$
(125) $\displaystyle\times\exp\left[i\int
dt\left[\widehat{x}(t)\left(\dot{x}(t)-ax(t)-g[M(t)]-\xi(t)-h_{i}(t)\right)\right]\right]$
$\displaystyle\times\exp\left[-i\int
dt\widehat{M}(t)x(t)\right]\exp\left[i\int dt~{}x(t)\psi_{i}(t)\right],$

with

$\displaystyle Z_{1}(\underline{\psi}_{i})$ $\displaystyle=$
$\displaystyle\int D\underline{\xi}{\cal D}\underline{x}{\cal
D}\widehat{}\underline{x}~{}p_{0}[x(0]\,p(\underline{\xi})~{}\exp\left[i\int
dt\left[\widehat{x}(t)\left(\dot{x}(t)-ax(t)-g[M(t)]-\xi(t)-h_{i}(t)\right)\right]\right]$
(126) $\displaystyle\times\exp\left[-i\int
dt\widehat{M}(t)x(t)\right]\exp\left[i\int dt~{}x(t)\psi_{i}(t)\right].$

####  5.5.2 Physical meaning of the order parameter $M(t)$ at the saddle point

We pointed out earlier that the physical interpretation of the saddle-point
value of the order parameter $M(t)=\frac{1}{N}\sum_{i}x_{i}(t)$ is not
entirely clear. To understand this better we introduce

$m(t)=\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}\left\langle{x_{i}(t)}\right\rangle,$
(127)

where the average on the right is an average over realisations of the
stochastic process for the $x_{i}$ in Eq. (120). This means averaging over
realisations of the dynamic noise variables $\xi_{i}(t)$ and over the initial
conditions for the $x_{i}(0)$, if they are random. The object $m(t)$ has a
clear interpretation in the original problem.

We will now show that $m(t)$ and $M(t)$ are the same at the saddle point and
in the thermodynamic limit.

From Eq. (98) we have

$Z[\underline{{\mbox{\boldmath$\psi$}}}]=\int
D\underline{\bm{\xi}}D\underline{\mathbf{x}}~{}p(\underline{\bm{\xi}})\,p_{0}[\mathbf{x}(0)]\,\delta(\mbox{equations
of motion})e^{i\sum_{i}\int dt~{}x_{i}(t)\psi_{i}(t)}.$ (128)

This means that $m(t)$ can be expressed in the form

$m(t)=-i\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}\left.\frac{\delta
Z[\underline{{\mbox{\boldmath$\psi$}}}]}{\delta\psi_{i}(t)}\right|_{\underline{{\mbox{\boldmath$\psi$}}}=\underline{\mathbf{0}}}.$
(129)

One the other hand $Z[{\mbox{\boldmath$\underline{\psi}$}}]$ is given by the
expression in Eq. (121), so that

$\displaystyle m(t)$ $\displaystyle=$
$\displaystyle-i\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}\left.\frac{\delta
Z[\underline{{\mbox{\boldmath$\psi$}}}]}{\delta\psi_{i}(t)}\right|_{\underline{{\mbox{\boldmath$\psi$}}}=\underline{\mathbf{0}}}$
(130) $\displaystyle=$
$\displaystyle-i\lim_{N\to\infty}\left.\frac{1}{N}\sum_{i}\int{\cal
D}[\underline{M},\widehat{}\underline{M}]\left\\{\exp\left[N\left(\Psi[\underline{M},\widehat{}\underline{M}]+\Omega[\underline{M},\widehat{}\underline{M},\underline{{\mbox{\boldmath$\psi$}}}]\right)\right]\,\,N\frac{\delta\Omega[\underline{M},\widehat{}\underline{M},\underline{{\mbox{\boldmath$\psi$}}}]}{\delta\psi_{i}(t)}\right\\}\right|_{\underline{{\mbox{\boldmath$\psi$}}}=\underline{\mathbf{0}}}$
$\displaystyle=$
$\displaystyle-i\lim_{N\to\infty}\sum_{i}\left.\frac{\delta\Omega[\underline{M},\widehat{}\underline{M},\underline{{\mbox{\boldmath$\psi$}}}]}{\delta\psi_{i}(t)}\right|_{\underline{{\mbox{\boldmath$\psi$}}}=\underline{\mathbf{0}},\mbox{SP}}$
$\displaystyle=$
$\displaystyle\left\langle{x(t)}\right\rangle_{\Omega}\Big{|}_{\underline{{\mbox{\boldmath$\psi$}}}=\underline{\mathbf{0}},\mbox{SP}}.$

The subscript ‘SP’ indicates that the relevant quantity is to be evaluated at
the saddle point. In the third step we have used the rules of saddle-point
integration, in particular Eq. (46). In the last step we have used the
expression for $\Omega$ in Eq. (123), and the definition of
$\left\langle{\cdots}\right\rangle_{\Omega}$ in Eqs. (125) and (126).

From the second relation in Eq. (5.5.1) we know that
$M(t)=\left\langle{x(t)}\right\rangle_{\Omega}$ at the saddle point. We can
therefore conclude that $m(t)$ and $M(t)$ are equal to one another at the
saddle point and in the limit
$\underline{{\mbox{\boldmath$\psi$}}}\to\underline{\mathbf{0}}$.

####  5.5.3 Proper derivation of the fact that $\widehat{M}(t)=0$ at the
saddle point

The source field $\underline{{\mbox{\boldmath$\psi$}}}$ has fulfilled its
purpose, and throughout this section we will set the $\psi_{i}(t)$ to zero for
all $i$ and $t$.

We start by defining the object

$p(t)=i\lim_{N\to\infty}\frac{1}{N}\sum_{i}\frac{\delta}{\delta
h_{i}(t)}\left\langle{1}\right\rangle.$ (131)

The average $\left\langle{\cdots}\right\rangle$ on the right is again an
average over realisations of the stochastic process in Eq. (120). This process
in turn involves the external perturbation fields $h_{i}(t)$, i.e., the
average $\left\langle{\cdots}\right\rangle$ is different for different choices
of $\underline{\mathbf{h}}$. As such one can define derivatives of the type
$\frac{\delta}{\delta h_{i}(t^{\prime})}\left\langle{x_{i}(t)}\right\rangle$
for example. But of course, the average of the quantity ‘one’ over a physical
process is always one, $\left\langle{1}\right\rangle=1$. This is the case
here, no matter what the fields $\underline{h}_{i}$ are. Therefore, we
trivially have

$p(t)=0,$ (132)

for all $t$.

It may sound odd to introduce an object when we already know that it is zero.
The point is though that we will now show that
$p(t)=\left\langle{\widehat{x}(t)}\right\rangle_{\Omega}$ at the saddle point.
This then allows us to conclude that
$\left\langle{\widehat{x}(t)}\right\rangle_{\Omega}=0$ (always assuming
$\underline{{\mbox{\boldmath$\psi$}}}=\underline{\mathbf{0}}$ ). Showing that
$\left\langle{\widehat{x}(t)}\right\rangle_{\Omega}=0$ was one of the problems
we had identified in Sec. 5.4.3.

We know that
$Z[{\mbox{\boldmath$\underline{\psi}$}}=\underline{\mathbf{0}}]=1$, no matter
what form the perturbation fields $h_{i}(t)$ take. Therefore
$p(t)=i\lim_{N\to\infty}\frac{1}{N}\sum_{i}\frac{\delta}{\delta
h_{i}(t)}Z[{\mbox{\boldmath$\underline{\psi}$}}=\underline{\mathbf{0}}]$
(simply because the objects on both sides of this equation are zero).

Using the expression for $Z[{\mbox{\boldmath$\underline{\psi}$}}]$ in Eq.
(121), we have

$\displaystyle p(t)$ $\displaystyle=$ $\displaystyle
i\lim_{N\to\infty}\left.\frac{1}{N}\sum_{i}\int{\cal
D}[\underline{M},\widehat{}\underline{M}]\left\\{\exp\left(N\left(\Psi[\underline{M},\widehat{}\underline{M}]+\Omega[\underline{M},\widehat{}\underline{M},\underline{{\mbox{\boldmath$\psi$}}}]\right)\right)\times
N\frac{\delta\Omega[\underline{M},\widehat{}\underline{M},\underline{{\mbox{\boldmath$\psi$}}}]}{\delta
h_{i}(t)}\right\\}\right|_{\underline{{\mbox{\boldmath$\psi$}}}=\underline{\mathbf{0}}}$
(133) $\displaystyle=$ $\displaystyle
i\lim_{N\to\infty}\sum_{i}\left.\frac{\delta\Omega[\underline{M},\widehat{}\underline{M},\underline{{\mbox{\boldmath$\psi$}}}]}{\delta
h_{i}(t)}\right|_{\underline{{\mbox{\boldmath$\psi$}}}=\underline{\mathbf{0}},\mbox{\footnotesize
SP}}$ $\displaystyle=$
$\displaystyle\left.\left\langle{\widehat{x}(t)}\right\rangle_{\Omega}\right|_{\underline{{\mbox{\boldmath$\psi$}}}=\underline{\mathbf{0}},\mbox{\footnotesize
SP}}.$

Given that we already know that $p(t)=0$, we have thus shown that
$\left\langle{\widehat{x}(t)}\right\rangle_{\Omega}=0$ at the saddle point and
in absence of the source field $\underline{{\mbox{\boldmath$\psi$}}}$. Using
the first relation in Eq. (5.5.1) this then also implies that
$\widehat{M}(t)=0$ for all $t$.

####  5.5.4 Single effective particle measure and effective process

We now set the source and perturbation fields to zero
($h_{i}(t)=0,\psi_{i}(t)=0$ for all $i,t$), as they are no longer needed. We
also operate at the saddle point. We can then use $\widehat{M}(t)=0$ for all
$t$ in Eqs. (125) and (126). Instead of
$\left\langle{\cdots}\right\rangle_{\Omega}$ we write
$\left\langle{\cdots}\right\rangle_{\star}$.

Noting that all terms in the sum over $i$ in Eq. (125) become the same, and
that $Z_{1}[\underline{\psi}_{i}=\underline{0}]=1$, we then have

$\displaystyle\left\langle{f(\underline{x},\widehat{}\underline{x})}\right\rangle_{\star}$
$\displaystyle=$ $\displaystyle\int D\underline{\xi}{\cal D}\underline{x}{\cal
D}\widehat{}\underline{x}~{}p_{0}[x(0]\,p(\underline{\xi})~{}f(\underline{x},\widehat{}\underline{x})$
(134) $\displaystyle\times\exp\left[i\int
dt\left[\widehat{x}(t)\left(\dot{x}(t)-ax(t)-g[M(t)]-\xi(t)\right)\right]\right],$

together with the following self-consistency relation [resulting from the
second relation in Eqs. (5.5.1)],

$M(t)=\left\langle{x(t)}\right\rangle_{\star}.$ (135)

The integral over the conjugate variables in Eq. (134) can, in principle, be
undone to produce delta functions of the type
$\delta[\dot{x}(t)-ax(t)-g[M(t)]-\xi(t)]$. Therefore,
$\left\langle{\cdots}\right\rangle_{\star}$ describes an average over
realisations of the process

$\dot{x}=ax+g(M)+\xi,$ (136)

which is what we wanted to show.

###  5.6 Summary of the main steps of the generating functional analysis

We briefly summarise the main steps of the generating functional analysis of
the toy model, as also illustrated in Fig. 2.

The main steps are:

  1. 1.

We set up the generating functional for the problem [Eq. (101)].

  2. 2.

We then introduced the macroscopic order parameters $\underline{M}$ and
$\widehat{}\underline{M}$ [Eqs (103, 104)].

  3. 3.

We showed that the generating functional can be written as an integral over
$\underline{M}$ and $\widehat{}\underline{M}$, and that this integral is in
saddle-point form in the limit $N\to\infty$ [Eq. (121)].

  4. 4.

We carried out the saddle-point integration, and derived the saddle-point
equations for the order parameters [Eqs. (5.5.1)].

  5. 5.

We showed that the conjugate order parameter vanishes, i.e.,
$\widehat{M}(t)=0$ for all $t$ [Sec. 5.5.3]. Alternatively, this can be
postulated and be justified retrospectively (Secs. 5.4.2 and 5.4.3).

  6. 6.

We showed that the remaining order parameter, $M(t)$, can be written as the
average over an effective single-particle process [Eq. (135) together with Eq.
(134)].

  7. 7.

The resulting dynamic mean field-theory thus consists of this effective
process [Eq. (136)] and an equation for the order parameter $M(t)$ [Eq.
(135)], involving an average over realisations of the effective process. This
combination is to be solved self-consistently.

In the toy problem the simple ordinary differential equation (119) can be
obtained for $M(t)$ from the effective process. This is a consequence of the
fact that the only potentially nonlinear term (nonlinear in the $x_{i}$) on
the right-hand side of the initial problem [Eq. (93)] can be expressed as a
function of $M$ (this is the term $g[M(t)]$). We would not be able to find
such simple closed equation for $M$ if the initial problem had been, say, of
the type
$\dot{x}_{i}=ax_{i}^{2}+g\left(\frac{1}{N}\sum_{j}x_{j}\right)+\xi_{i}(t)$
[the effective single-particle process would then take the form
$\dot{x}=ax^{2}+g(M)+\xi$.]

Figure 2: Illustration of the main steps of the generating functional analysis
for the toy model.

##  6 Generating functional analysis for random Lotka–Volterra equations I:
Setup, disorder average and saddle-point equations

###  6.1 Overview of this section

We now turn to the Lotka–Volterra problem. In Sec. 6.2 we summarise the setup
of the model, and briefly describe how realisations of the random interaction
coefficients can be generated numerically.

We then present the initial steps of the generating functional calculation,
namely:

  1. 1.

Setting up the generating functional for the problem [Eq. (141)].

  2. 2.

Compute the disorder average of the generating functional (Sec. 6.4).

  3. 3.

Introduction of the macroscopic order parameters [Sec. 6.5, in particular Eq.
(155)].

  4. 4.

Write the disorder-averaged generating functional as a saddle-point integral
over the order parameters and their conjugates [Eq. (159)].

  5. 5.

Carry out the saddle-point integration, and derive the saddle-point equations
for the order parameters [Sec. 6.6, in particular Eqs. (164) and (165)].

Of course, these steps are similar to steps 1-4 in the calculation of the toy
model, summarised in Sec. 5.6. The main addition is the new step 2, the
disorder average. This was not necessary earlier, as there was no disorder in
the toy problem. Also, there are now significantly more order parameters than
in the toy problem.

Once step 5 above has been reached (saddle-point equations for the order
parameters in the Lotka–Volterra problem), there is again a simpler ’vanilla’
method of deriving the dynamical mean-field theory, or a more robust, but
longer method. These are described in Secs. 7 and 8 respectively, and are more
involved than their counterparts for the toy model. Both methods lead to the
same effective single-species process (the dynamical mean-field theory for the
Lotka–Volterra problem).

###  6.2 Setup

####  6.2.1 Model definitions

The calculation follows [3], and is based on the principles of [50, 5, 51, 9].
It was originally developed in the context of random replicator models in
[13], and then used for example also in [14, 52, 15, 16, 17, 18, 19, 20, 21].

We start from the Lotka–Volterra equations

$\dot{x}_{i}=x_{i}\left(1-x_{i}+\sum_{j}\alpha_{ij}x_{j}\right),$ (137)

with Gaussian random interaction coefficients $\alpha_{ij}$. We set
$\alpha_{ii}=0$, and as in [3] we assume that for each pair $i<j$,

$\displaystyle\overline{a_{ij}}=\overline{\alpha_{ji}}$ $\displaystyle=$
$\displaystyle\frac{\mu}{N},$
$\displaystyle\mbox{Var}(\alpha_{ij})=\mbox{Var}(\alpha_{ji})$
$\displaystyle=$ $\displaystyle\frac{\sigma^{2}}{N},$
$\displaystyle\mbox{Cov}(\alpha_{ij},\alpha_{ji})$ $\displaystyle=$
$\displaystyle\Gamma.$ (138)

We have written $\overline{\cdots}$ for the average over the random couplings.
$\mbox{Var}(\alpha)=\overline{\alpha^{2}}-(\overline{\alpha})^{2}$ stands for
the variance, and
$\mbox{Cov}(\alpha,\beta)=\left[\overline{\alpha\beta}-\overline{\alpha}\overline{\beta}\right]/\sqrt{\mbox{Var}(\alpha)\mbox{Var}(\beta)}$
is the covariance. The quantities $\mu,\sigma^{2}$ and $\Gamma$ are the
parameters of the model. (We note that the correlation parameter is called
$\gamma$ in [3]. However, we here prefer to use $\Gamma$ in-line with later
work, including [17].) The parameter $\mu$ can take any real value, and
$\Gamma$ is restricted to the interval $[-1,1]$. We assume that elements
$\alpha_{ij}$ and $\alpha_{k\ell}$ are uncorrelated, unless the $i,j,k,\ell$
are such that the two elements are identical ($i=k,j=\ell$), or diagonally
opposed to each other in the matrix ($i=\ell,j=k$).

####  6.2.2 Numerical generation of random matrix with correlations between
diagonally opposed elements

A common question I receive from students is how to generate instances of the
random matrix $\underline{\underline{\alpha}}$ with statistics as in Eqs.
(138). Generally, a set of correlated Gaussian random numbers with a given
covariance matrix can be generated using a Cholesky decomposition of the
correlation matrix. If you have not heard this, then this is probably a term
worth looking up,

We here only describe the construction for our specific model. A matrix
$\underline{\underline{\alpha}}$ with the above statistics can be generated as
follows (for a given finite $N$). For each pair $i<j$ proceed as follows:

  1. 1.

Generate two independent random Gaussian variables $u$ and $v$, each with mean
zero and variance one. The variables $u$ and $v$ for one pair $(i,j)$ are
independent from those of any other pair.

  2. 2.

Then set $\alpha_{ij}=\frac{\mu}{N}+\frac{\sigma}{\sqrt{N}}u$ and
$\alpha_{ji}=\frac{\mu}{N}+\frac{\sigma}{\sqrt{N}}\left(\Gamma
u+\sqrt{1-\Gamma^{2}}\right)v$.

One can then directly check that the resulting $\\{\alpha_{ij}\\}$ have the
properties in Eqs. (138). For example,

$\displaystyle\overline{(\alpha_{ji}-\mu/N)^{2}}$ $\displaystyle=$
$\displaystyle\frac{\sigma^{2}}{N}\left[\Gamma^{2}\underbrace{\overline{u^{2}}}_{=1}+(1-\Gamma^{2})\underbrace{\overline{v^{2}}}_{=1}+\Gamma\sqrt{1-\Gamma^{2}}\underbrace{\overline{uv}}_{=0}\right]$
(139) $\displaystyle=$ $\displaystyle\frac{\sigma^{2}}{N}.$

###  6.3 Generating functional

There is a clean way of implementing the generating functional calculation and
a slightly ‘dirty’ approach. (I stress that this distinction is different from
the distinction between the ‘vanilla’ and proper treatments of the toy
problem.) The advantage of the clean implementation is, well, that it is
clean. The advantage of the dirty approach is that it is perhaps easier to
follow for someone who isn’t familiar with dynamic mean field theory for
disordered systems. I here take the ‘dirty’ route, noting that the sloppiness
of this approach is only in the first line, slightly re-phrasing the setup,
and not in the actual generating-functional calculation.

The sloppy step consists of dividing both sides of Eq. (137) by $x_{i}$. That
is, we start from

$\frac{\dot{x}_{i}}{x_{i}}=1-x_{i}+\sum_{j\neq i}\alpha_{ij}x_{j}.$ (140)

Of course, this is only allowed if $x_{i}$ remains non-zero. This is
technically not true, because (as we will see) some species die out in the
long run. Nonetheless, we’ll proceed starting from Eq. (140). The clean method
is described in Appendix A. Ultimately, both routes lead to the same result,
justifying the sloppiness which we exercise purely for pedagogic reasons.

For a fixed realisation of the disorder, the dynamical generating functional
of the process is then

$\displaystyle Z[{\mbox{\boldmath$\psi$}}]$ $\displaystyle=$
$\displaystyle\int{\cal D}\underline{\mathbf{x}}\,{\cal
D}\widehat{}\underline{\mathbf{x}}\,p_{0}[\mathbf{x}(0)]\,\exp\left(i\sum_{i}\int
dt~{}x_{i}(t)\psi_{i}(t)\right)$ (141)
$\displaystyle\times\exp\Bigg{(}i\sum_{i}\int
dt\Bigg{[}\widehat{x}_{i}(t)\Bigg{(}\frac{\dot{x}_{i}(t)}{x_{i}(t)}-[1-x_{i}(t)+\sum_{j\neq
i}\alpha_{ij}x_{j}(t)+h_{i}(t)]\Bigg{)}\Bigg{]}\Bigg{)},$

where we have enforced the equations of motion via suitable delta-functions,
written in their exponential representation, similar to the way we did this in
Sec. 4.2. We have also added a perturbation field $\underline{\mathbf{h}}$. As
in the toy problem we note that this field is not part of the actual model,
but only a mathematical device used to generate response functions. We will
set $\underline{\mathbf{h}}$ to zero at the end of the calculation.

###  6.4 Carrying out the disorder average

Next, we look at the terms containing the disorder (the $\\{a_{ij}\\}$), and
perform the Gaussian average over these random variables. For any pair $i<j$
of species we can write

$\alpha_{ij}=\frac{\mu}{N}+\frac{\sigma}{\sqrt{N}}z_{ij},~{}~{}~{}\alpha_{ji}=\frac{\mu}{N}+\frac{\sigma}{\sqrt{N}}z_{ji},$
(142)

where $z_{ij}$ and $z_{ji}$ are drawn from a Gaussian distribution with
$\overline{z_{ij}}=0$, $\overline{z_{ij}^{2}}=1$, and
$\overline{z_{ij}z_{ji}}=\Gamma$.

The term in the generating functional [Eq. (141)] containing the disorder is

$X\equiv\exp\left(-i\sum_{i\neq j}\int
dt\,\widehat{x}_{i}(t)\alpha_{ij}x_{j}(t)\right).$ (143)

First, we have, using Eqs. (142),

$\displaystyle\overline{X}$ $\displaystyle=$
$\displaystyle\overline{\exp\left(-i\sum_{i\neq j}\int
dt\,\widehat{x}_{i}(t)\alpha_{ij}x_{j}(t)\right)}$ (144) $\displaystyle=$
$\displaystyle\exp\left(-i\frac{\mu}{N}\sum_{i\neq j}\int
dt\,\widehat{x}_{i}(t)x_{j}(t)\right)\times\overline{\exp\left(-i\frac{\sigma}{\sqrt{N}}\sum_{i\neq
j}\int dt\,\widehat{x}_{i}(t)z_{ij}x_{j}(t)\right)}.$

The first factor in this does not contain any disorder. We therefore look at
the second factor, which we’ll call $Y$.

We can write

$\displaystyle Y$ $\displaystyle\equiv$
$\displaystyle\exp\left(-i\frac{\sigma}{\sqrt{N}}\sum_{i\neq j}\int
dt\,\widehat{x}_{i}(t)z_{ij}x_{j}(t)\right)$ (145) $\displaystyle=$
$\displaystyle\prod_{i<j}\exp\left(-i\frac{\sigma}{\sqrt{N}}\int
dt\,\\{\widehat{x}_{i}(t)z_{ij}x_{j}(t)+\widehat{x}_{j}(t)z_{ji}x_{i}(t)\\}\right).$

We have split this in this way, because $z_{ij}$ and $z_{ji}$ are correlated,
so we need to average over these two quantities together.

The factor

$Y_{ij}\equiv\exp\left(-i\frac{\sigma}{\sqrt{N}}\int
dt\,\\{\widehat{x}_{i}(t)z_{ij}x_{j}(t)+\widehat{x}_{j}(t)z_{ji}x_{i}(t)\\}\right)$
(146)

is of the form $\exp\left(z_{1}A_{ij}+z_{2}B_{ij}\right)$, with $z_{1}$,
$z_{2}$ Gaussian random variables, with mean zero, variance one, and
correlation $\overline{z_{1}z_{2}}=\Gamma$. $A_{ij}$ and $B_{ij}$ are
shorthands for

$\displaystyle A_{ij}$ $\displaystyle=$
$\displaystyle-i\frac{\sigma}{\sqrt{N}}\int dt\,\widehat{x}_{i}(t)x_{j}(t),$
$\displaystyle B_{ij}$ $\displaystyle=$
$\displaystyle-i\frac{\sigma}{\sqrt{N}}\int
dt\,\widehat{x}_{j}(t)x_{i}(t)~{}(=A_{ji}).$ (147)

Carrying out the average over the Gaussian variables $z_{1}$ and $z_{2}$ is
then an exercise in bivariate Gaussian integration. This can be done using the
expressions in Sec. 3.4.

We have to evaluate

$\displaystyle\overline{Y}_{ij}=\frac{1}{\sqrt{(2\pi)^{2}\det\Sigma}}\int
dz_{1}\int
dz_{2}\exp\left(-\frac{1}{2}\mathbf{z}^{T}\Sigma^{-1}\mathbf{z}\right)\exp\left(z_{1}A_{ij}+z_{2}B_{ij}\right),$
(148)

where $\mathbf{z}=(z_{1},z_{2})$ and where the $2\times 2$ matrix $\Sigma$ has
entries $\Sigma_{ab}=\overline{z_{a}z_{b}}$ for $a,b,\in\\{1,2\\}$, i.e.,
$\Sigma=\left(\begin{array}[]{cc}1&\Gamma\\\ \Gamma&1\end{array}\right)$.

To evaluate the integral we use Eq. (30). The result is

$\displaystyle\overline{Y}_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{(2\pi)^{2}\det\Sigma}}\int dz_{1}\int
dz_{2}\exp\left(-\frac{1}{2}\mathbf{z}^{T}\Sigma^{-1}\mathbf{z}\right)\exp\left(z_{1}A_{ij}+z_{2}B_{ij}\right)$
(149) $\displaystyle=$ $\displaystyle\exp\left(\frac{1}{2}(A_{ij}^{2}+2\Gamma
A_{ij}B_{ji}+B_{ij}^{2})\right).$

We note that the relation in Eq. (149) is an exact identity, nothing needs to
be assumed about the scaling of $A_{ij}$ and $B_{ji}$ with $N$.

Aside (alternative way of carrying out the disorder average, assuming $N\gg
1$):
This method relies on $A_{ij}$ and $B_{ij}$ scaling as $N^{-1/2}$, see Eqs.
(6.4). Eventually, we are interested in the limit of large $N$. So we expand
each factor $Y_{ij}$ in Eq. (146) in powers of $1/\sqrt{N}$, or equivalently,
powers of $A_{ij}$ and $B_{ij}$. One gets

$\displaystyle\exp(z_{1}A_{ij}+z_{2}B_{ij})$ $\displaystyle=$ $\displaystyle
1+\left(z_{1}A_{ij}+z_{2}B_{ij}\right)+\frac{1}{2}(z_{1}A_{ij}+z_{2}B_{ij})^{2}+{\cal
O}(1/N^{3/2})$ $\displaystyle=$ $\displaystyle
1+z_{1}A_{ij}+z_{2}B_{ij}+\frac{1}{2}(z_{1}^{2}A_{ij}^{2}+2z_{1}z_{2}A_{ij}B_{ij}+z_{2}^{2}B_{ij}^{2})+{\cal
O}(1/N^{3/2}).$

Next we average over the disorder (keeping in mind that $z_{1}$ and $z_{2}$
each have mean zero, variance one, and that their correlation is
$\overline{z_{1}z_{2}}=\Gamma$):

$\displaystyle\overline{\exp(z_{1}A_{ij}+z_{2}B_{ij}})$ $\displaystyle=$
$\displaystyle 1+\frac{1}{2}(A_{ij}^{2}+2\Gamma A_{ij}B_{ij}+B_{ij}^{2})+{\cal
O}(1/N^{2}).$ (151)

[After averaging the correction term in the exponential is now of order
$N^{-2}$. This is because terms of third order in $z_{1}$ and $z_{2}$ in the
expansion in Eq. (6.4) average to zero.] Next we re-exponentiate and find

$\displaystyle\overline{Y}_{ij}$ $\displaystyle=$ $\displaystyle
1+\frac{1}{2}(A_{ij}^{2}+2\Gamma A_{ij}B_{ij}+B_{ij}^{2})+{\cal O}(1/N^{2})$
(152) $\displaystyle=$ $\displaystyle\exp\left(\frac{1}{2}(A_{ij}^{2}+2\Gamma
A_{ij}B_{ij}+B_{ij}^{2})+{\cal O}(1/N^{2})\right).$

To leading order in $1/N$ the expression in Eq. (152) is the same as the exact
result in Eq. (149). However, the derivation of Eq. (152) as just presented is
based on the assumption $N\gg 1$.

###  6.5 Introduction of macroscopic order parameters

Using Eqs. (144) and (149) we have arrived at

$\displaystyle\overline{X}$ $\displaystyle=$
$\displaystyle\overline{\exp\left(-i\sum_{i\neq j}\int
dt\,\widehat{x}_{i}(t)\alpha_{ij}x_{j}(t)\right)}$ (153) $\displaystyle=$
$\displaystyle\exp\left(-i\frac{\mu}{N}\sum_{i\neq j}\int
dt\,\widehat{x}_{i}(t)x_{j}(t)\right)\times\exp\left(\frac{1}{2}\sum_{i<j}(A_{ij}^{2}+2\Gamma
A_{ij}A_{ji}+A_{ji}^{2})\right)$ $\displaystyle=$
$\displaystyle\exp\left(-i\frac{\mu}{N}\sum_{i\neq j}\int
dt\,\widehat{x}_{i}(t)x_{j}(t)\right)\times\exp\left(\frac{1}{2}\sum_{i\neq
j}(A_{ij}^{2}+\Gamma A_{ij}A_{ji})\right),$

with $A_{ij}=-i\frac{\sigma}{\sqrt{N}}\int dt\,\widehat{x}_{i}(t)x_{j}(t)$. We
note that

$\displaystyle A_{ij}^{2}$ $\displaystyle=$
$\displaystyle-\frac{\sigma^{2}}{N}\int
dt\,dt^{\prime}\,\widehat{x}_{i}(t)x_{j}(t)\widehat{x}_{i}(t^{\prime})x_{j}(t^{\prime}),$
$\displaystyle A_{ij}A_{ji}$ $\displaystyle=$
$\displaystyle-\frac{\sigma^{2}}{N}\int
dt\,dt^{\prime}\,\widehat{x}_{i}(t)x_{j}(t)\widehat{x}_{j}(t^{\prime})x_{i}(t^{\prime}).$
(154)

Next we introduce the following short-hands,

$\displaystyle M(t)=\frac{1}{N}\sum_{i}x_{i}(t),$ $\displaystyle
P(t)=\frac{1}{N}\sum_{i}\widehat{x}_{i}(t),$ $\displaystyle
C(t,t^{\prime})=\frac{1}{N}\sum_{i}x_{i}(t)x_{i}(t^{\prime}),$ $\displaystyle
K(t,t^{\prime})=\frac{1}{N}\sum_{i}x_{i}(t)\widehat{x}_{i}(t^{\prime}),$
$\displaystyle
L(t,t^{\prime})=\frac{1}{N}\sum_{i}\widehat{x}_{i}(t)\widehat{x}_{i}(t^{\prime}).$
(155)

With these definitions we can write

$\displaystyle\overline{\exp\left(-i\sum_{i\neq j}\int
dt\,\widehat{x}_{i}(t)\alpha_{ij}x_{j}(t)\right)}$ (156) $\displaystyle=$
$\displaystyle\exp\left(-i\mu N\int dt\,P(t)M(t)\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}\left.-\frac{1}{2}N\sigma^{2}\int
dt~{}dt^{\prime}\left[L(t,t^{\prime})C(t,t^{\prime})+\Gamma
K(t,t^{\prime})K(t^{\prime},t)\right]+{\cal O}(N^{0})\right).$

As indicated by the term ${\cal O}(N^{0})$, we have left out sub-leading
contributions in $N$ (i.e, terms of order $N^{0}$ or lower). For example we
had $-i\frac{\mu}{N}\sum_{i\neq j}\int dt\,\widehat{x}_{i}(t)x_{j}(t)$ in the
exponential in Eq. (153), and we have now replaced this with $-\mu N\int
dt\,P(t)M(t)=-i\frac{\mu}{N}\sum_{i,j}\int dt\widehat{x}_{i}(t)x_{j}(t)$.
Technically, this introduces an additional contribution
$-i\frac{\mu}{N}\sum_{i}\int dt~{}\widehat{x}_{i}(t)x_{i}(t)$. However, this
term is of order $N^{0}$, and therefore a sub-leading contribution (in powers
of $1/N$). We can ignore this, because we are eventually going to take the
limit $N\to\infty$. We only keep terms in the exponent that are of order $N$.

The order parameters can formally be introduced into the generating functional
as delta-functions in their integral representation, e.g.

$\displaystyle 1$ $\displaystyle=$
$\displaystyle\int\prod_{t,t^{\prime}}dC(t,t^{\prime})\delta\left(C(t,t^{\prime})-\frac{1}{N}\sum_{i}x_{i}(t)x_{i}(t^{\prime})\right)$
(157) $\displaystyle=$ $\displaystyle\int{\cal
D}[\underline{\underline{C}},\widehat{}\underline{\underline{C}}]~{}\exp\left(iN\int
dt~{}dt^{\prime}\widehat{C}(t,t^{\prime})\left(C(t,t^{\prime})-N^{-1}\sum_{i}x_{i}(t)x_{i}(t^{\prime})\right)\right),$

with

${\cal
D}[\underline{\underline{C}},\widehat{}\underline{\underline{C}}]=\prod_{t,t^{\prime}}\frac{dC(t,t^{\prime})d\widehat{C}(t,t^{\prime})}{2\pi/N}.$
(158)

We also insert similar expressions for the other order parameters.

The disorder-averaged generating functional can be written as follows

$\overline{Z}[\underline{{\mbox{\boldmath$\psi$}}}]=\int{\cal
D}[\underline{M},\widehat{}\underline{M}]{\cal
D}[\underline{P},\widehat{}\underline{P}]{\cal
D}[\underline{\underline{C}},\widehat{}\underline{\underline{C}}]{\cal
D}[\underline{\underline{K}},\widehat{}\underline{\underline{K}}]{\cal
D}[\underline{\underline{L}},\widehat{}\underline{\underline{L}}]\,\exp\left(N\left[\Psi+\Phi+\Omega+{\cal
O}(N^{-1})\right]\right).$ (159)

The term

$\displaystyle\Psi$ $\displaystyle=$ $\displaystyle i\int
dt~{}[\widehat{M}(t)M(t)+\widehat{P}(t)P(t)]$ (160) $\displaystyle+i\int
dt~{}dt^{\prime}\left[\widehat{C}(t,t^{\prime})C(t,t^{\prime})+\widehat{K}(t,t^{\prime})K(t,t^{\prime})+\widehat{L}(t,t^{\prime})L(t,t^{\prime})\right]$

results from the introduction of the macroscopic order parameters. The
contribution

$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\sigma^{2}\int
dt~{}dt^{\prime}\left[L(t,t^{\prime})C(t,t^{\prime})+\Gamma
K(t,t^{\prime})K(t^{\prime},t)\right]-i\mu\int dt~{}M(t)P(t)$ (161)

comes from the disorder average, and $\Omega$ describes the details of the
microscopic time evolution

$\displaystyle\Omega$ $\displaystyle=$ $\displaystyle
N^{-1}\sum_{i}\ln\bigg{[}\int{\cal D}\underline{x}_{i}{\cal
D}\widehat{}\underline{x}_{i}\,p_{0}^{(i)}[x_{i}(0)]\exp\left(i\int
dt~{}\psi_{i}(t)x_{i}(t)\right)$ (162) $\displaystyle\times\exp\left(i\int
dt~{}\widehat{x}_{i}(t)\left(\frac{\dot{x}_{i}(t)}{x_{i}(t)}-[1-x_{i}(t)]-h_{i}(t)\right)\right)$
$\displaystyle\times\exp\left(-i\int
dt~{}dt^{\prime}\left[\widehat{C}(t,t^{\prime})x_{i}(t)x_{i}(t^{\prime})+\widehat{L}(t,t^{\prime})\widehat{x}_{i}(t)\widehat{x}_{i}(t^{\prime})+\widehat{K}(t,t^{\prime})x_{i}(t)\widehat{x}_{i}(t^{\prime})\right]\right)$
$\displaystyle\times\exp\left(-i\int
dt~{}[\widehat{M}(t)x_{i}(t)+\widehat{P}(t)\widehat{x}_{i}(t)]\right)\bigg{]}.$

The quantity $p_{0}^{(i)}[\cdot]$ describes the distribution from which the
initial value of $x_{i}$ (at time $t=0$) is drawn. In writing down Eq. (162)
we have assumed that the joint distribution of all $x_{i}(0)$ ($i=1,\dots,N)$
factorises, i.e., $p_{0}[\mathbf{x}(0)]=\prod_{i}p_{0}^{(i)}[x_{i}(0)]$. We’ll
now go even further and assume that the $x_{i}(0)$ are independent and
identically distributed (iid), that is, we write
$p_{0}^{(i)}[x_{i}(0)]=p_{0}[x_{i}(0)]$ for all $i$.

We note that the $\\{x_{i}(t),\widehat{x}_{i}(t)\\}$ in the expression for
$\Omega$ are dummy variables to be integrated over. We can therefore re-write
$\Omega$ in the following form

$\displaystyle\Omega$ $\displaystyle=$ $\displaystyle
N^{-1}\sum_{i}\ln\bigg{[}\int{\cal D}\underline{x}{\cal
D}\widehat{}\underline{x}\,p_{0}[x(0)]\exp\left(i\int
dt~{}\psi_{i}(t)x(t)\right)$ (163) $\displaystyle\times\exp\left(i\int
dt~{}\widehat{x}(t)\left(\frac{\dot{x}(t)}{x(t)}-[1-x(t)]-h_{i}(t)\right)\right)$
$\displaystyle\times\exp\left(-i\int
dt~{}dt^{\prime}\left[\widehat{C}(t,t^{\prime})x(t)x(t^{\prime})+\widehat{L}(t,t^{\prime})\widehat{x}(t)\widehat{x}(t^{\prime})+\widehat{K}(t,t^{\prime})x(t)\widehat{x}(t^{\prime})\right]\right)$
$\displaystyle\times\exp\left(-i\int
dt~{}[\widehat{M}(t)x(t)+\widehat{P}(t)\widehat{x}(t)]\right)\bigg{]}.$

The object inside the logarithm on the right contains $\psi_{i}(\cdot)$ and
$h_{i}(\cdot)$. Therefore, the different terms in the sum over $i$ on the
right are not identical at this point.

###  6.6 Saddle-point analysis

####  6.6.1 Saddle-point equations for the macroscopic order parameters

We next use the saddle-point method (Sec. 3.5) to carry out the integrals in
Eq. (159). This is valid in the limit $N\to\infty$, and amounts to finding the
extrema of the term in the exponent.

Setting the variation with respect to components of the the integration
variables
$\underline{M},\underline{P},\underline{\underline{C}},\underline{\underline{K}}$
and $\underline{\underline{L}}$ to zero gives, respectively,

$\displaystyle i\widehat{M}(t)$ $\displaystyle=$ $\displaystyle i\mu P(t),$
$\displaystyle i\widehat{P}(t)$ $\displaystyle=$ $\displaystyle i\mu M(t),$
$\displaystyle i\widehat{C}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sigma^{2}L(t,t^{\prime}),$ $\displaystyle
i\widehat{K}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle\Gamma\sigma^{2}K(t^{\prime},t),$ $\displaystyle
i\widehat{L}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sigma^{2}C(t,t^{\prime}).$ (164)

Next we extremise with respect to components of
$\widehat{}\underline{M},\widehat{}\underline{P},\widehat{}\underline{\underline{C}},\widehat{}\underline{\underline{K}},\widehat{}\underline{\underline{L}}$.
We find

$\displaystyle M(t)$ $\displaystyle=$ $\displaystyle
i\lim_{N\to\infty}\frac{\delta\Omega}{\delta\widehat{M}(t)},$ $\displaystyle
P(t)$ $\displaystyle=$ $\displaystyle
i\lim_{N\to\infty}\frac{\delta\Omega}{\delta\widehat{P}(t)},$ $\displaystyle
C(t,t^{\prime})$ $\displaystyle=$ $\displaystyle
i\lim_{N\to\infty}\frac{\delta\Omega}{\delta\widehat{C}(t,t^{\prime})},$
$\displaystyle K(t,t^{\prime})$ $\displaystyle=$ $\displaystyle
i\lim_{N\to\infty}\frac{\delta\Omega}{\delta\widehat{K}(t,t^{\prime})},$
$\displaystyle L(t,t^{\prime})$ $\displaystyle=$ $\displaystyle
i\lim_{N\to\infty}\frac{\delta\Omega}{\delta\widehat{L}(t,t^{\prime})}.$ (165)

The object $\Omega$ in Eq. (163) is of the form
$\Omega=\frac{1}{N}\sum_{i}\ln[\mbox{complicated thing}_{i}]$, where the
subscript $i$ indicates that the object ‘complicated thing’ is different for
each $i$ (because of the $\underline{\psi}_{i}$ and $\underline{h}_{i}$ as
mentioned above). Derivatives of $\Omega$ are then of the form

$\displaystyle\frac{\delta\Omega}{\delta\mbox{something}}=\frac{1}{N}\sum_{i}\frac{1}{\mbox{complicated
thing}_{i}}\frac{\delta\mbox{complicated thing}_{i}}{\delta\mbox{something}}.$
(166)

From now on we’ll use more scientific notation and write
$Z_{1}[\underline{\psi}_{i},\underline{h}_{i}]$ instead of ‘$\mbox{complicated
thing}_{i}$’. That is, we define

$\displaystyle Z_{1}[\underline{\psi}_{i},\underline{h}_{i}]$
$\displaystyle\equiv$ $\displaystyle\int{\cal D}\underline{x}{\cal
D}\widehat{}\underline{x}\,p_{0}[x(0)]\exp\left(i\int
dt~{}\psi_{i}(t)x(t)\right)$ (167) $\displaystyle\times\exp\left(i\int
dt~{}\widehat{x}(t)\left(\frac{\dot{x}(t)}{x(t)}-[1-x(t)]-h_{i}(t)\right)\right)$
$\displaystyle\times\exp\left(-i\int
dt~{}dt^{\prime}\left[\widehat{C}(t,t^{\prime})x(t)x(t^{\prime})+\widehat{L}(t,t^{\prime})\widehat{x}(t)\widehat{x}(t^{\prime})+\widehat{K}(t,t^{\prime})x(t)\widehat{x}(t^{\prime})\right]\right)$
$\displaystyle\times\exp\left(-i\int
dt~{}[\widehat{M}(t)x(t)+\widehat{P}(t)\widehat{x}(t)]\right).$

Next, we note that taking a derivative of $Z_{1}$ with respect to one of the
hatted order parameters ‘brings down’ objects involving $x$ or $\widehat{x}$
in the integral. For example,

$\displaystyle\frac{\delta
Z_{1}[\underline{\psi}_{i},\underline{h}_{i}]}{\delta\widehat{C}(t_{1},t_{2})}$
$\displaystyle=$ $\displaystyle-i\int{\cal D}\underline{x}{\cal
D}\widehat{}\underline{x}\,p_{0}[x(0)]~{}x(t_{1})x(t_{2})~{}\exp\left(i\int
dt~{}\psi_{i}(t)x(t)\right)$ (168) $\displaystyle\times\exp\left(i\int
dt~{}\widehat{x}(t)\left(\frac{\dot{x}(t)}{x(t)}-[1-x(t)]-h_{i}(t)\right)\right)$
$\displaystyle\times\exp\left(-i\int
dt~{}dt^{\prime}\left[\widehat{C}(t,t^{\prime})x(t)x(t^{\prime})+\widehat{L}(t,t^{\prime})\widehat{x}(t)\widehat{x}(t^{\prime})+\widehat{K}(t,t^{\prime})x(t)\widehat{x}(t^{\prime})\right]\right)$
$\displaystyle\times\exp\left(-i\int
dt~{}[\widehat{M}(t)x(t)+\widehat{P}(t)\widehat{x}(t)]\right),$

where it is important to notice the factor $x(t_{1})x(t_{2})$ in the first
line.

With this, we have from Eqs. (165),

$\displaystyle M(t)$ $\displaystyle=$
$\displaystyle\lim_{N\to\infty}N^{-1}\sum_{i}\left\langle{x(t)}\right\rangle_{i},$
$\displaystyle P(t)$ $\displaystyle=$
$\displaystyle\lim_{N\to\infty}N^{-1}\sum_{i}\left\langle{\widehat{x}(t)}\right\rangle_{i},$
$\displaystyle C(t,t^{\prime})$ $\displaystyle=$
$\displaystyle\lim_{N\to\infty}N^{-1}\sum_{i}\left\langle{x(t)x(t^{\prime})}\right\rangle_{i},$
$\displaystyle K(t,t^{\prime})$ $\displaystyle=$
$\displaystyle\lim_{N\to\infty}N^{-1}\sum_{i}\left\langle{x(t)\widehat{x}(t^{\prime})}\right\rangle_{i},$
$\displaystyle L(t,t^{\prime})$ $\displaystyle=$
$\displaystyle\lim_{N\to\infty}N^{-1}\sum_{i}\left\langle{\widehat{x}(t)\widehat{x}(t^{\prime})}\right\rangle_{i},$
(169)

where
$\left\langle{g[\underline{x},\widehat{}\underline{x}]}\right\rangle_{i}$
stands for

$\displaystyle\left\langle{g[\underline{x},\widehat{}\underline{x}]}\right\rangle_{i}$
$\displaystyle=$
$\displaystyle\frac{1}{Z_{1}[\underline{\psi}_{i},\underline{h}_{i}]}\times\int{\cal
D}\underline{x}{\cal
D}\widehat{}\underline{x}\,p_{0}[x(0)]~{}g[\underline{x},\widehat{}\underline{x}]\exp\left(i\int
dt~{}\psi_{i}(t)x(t)\right)$ (170) $\displaystyle\times\exp\left(i\int
dt~{}\widehat{x}(t)\left(\frac{\dot{x}(t)}{x(t)}-[1-x(t)]-h_{i}(t)\right)\right)$
$\displaystyle\times\exp\left(-i\int
dt~{}dt^{\prime}\left[\widehat{C}(t,t^{\prime})x(t)x(t^{\prime})+\widehat{L}(t,t^{\prime})\widehat{x}(t)\widehat{x}(t^{\prime})+\widehat{K}(t,t^{\prime})x(t)\widehat{x}(t^{\prime})\right]\right)$
$\displaystyle\times\exp\left(-i\int
dt~{}[\widehat{M}(t)x(t)+\widehat{P}(t)\widehat{x}(t)]\right).$

For convenience we repeat the expression for
$Z_{1}[\underline{\psi}_{i},\underline{h}_{i}]$,

$\displaystyle Z_{1}[\underline{\psi}_{i},\underline{h}_{i}]$ $\displaystyle=$
$\displaystyle\int{\cal D}\underline{x}{\cal
D}\widehat{}\underline{x}\,p_{0}[x(0)]~{}\exp\left(i\int
dt~{}\psi_{i}(t)x(t)\right)$ (171) $\displaystyle\times\exp\left(i\int
dt~{}\widehat{x}(t)\left(\frac{\dot{x}(t)}{x(t)}-[1-x(t)]-h_{i}(t)\right)\right)$
$\displaystyle\times\exp\left(-i\int
dt~{}dt^{\prime}\left[\widehat{C}(t,t^{\prime})x(t)x(t^{\prime})+\widehat{L}(t,t^{\prime})\widehat{x}(t)\widehat{x}(t^{\prime})+\widehat{K}(t,t^{\prime})x(t)\widehat{x}(t^{\prime})\right]\right)$
$\displaystyle\times\exp\left(-i\int
dt~{}[\widehat{M}(t)x(t)+\widehat{P}(t)\widehat{x}(t)]\right).$

####  6.6.2 Next steps

The ultimate goal of our generating-functional calculation for the
Lotka–Volterra system is to derive a dynamic mean-field theory, that is a
stochastic process for an effective single species. Our next step in this
direction is to simplify the expressions in Eqs. (170) and (171). For example,
we will again show that $P(t)=0$ and $\widehat{M}(t)=0$ for all $t$.

As for the toy model we first present a simplified and perhaps not fully
satisfactory derivation (Sec. 7). However, the general idea is right, and most
importantly, the effective process that comes out of this is right. A more
complete derivation is given in Sec. 8.

##  7 Generating functional analysis for random Lotka–Volterra equations II:
‘Vanilla’ derivation of the effective single-species process

###  7.1 Simplification of saddle-point equations

We start from Eqs. (170) and (171). Our first move is to focus on the
situation in which $\psi_{i}(t)=\psi(t)$ for all $i$. We also set
$h_{i}(t)=h(t)$. There is nothing dubious about this, it is simply a
restriction to a special case. We then find that
$\left\langle{\cdots}\right\rangle_{i}$ in Eq. (170) carries no
$i$-dependence, and we write $\left\langle{\cdots}\right\rangle_{\star}$
instead.

The dodgy step comes now. Both $\underline{P}$ and $\underline{\underline{L}}$
are averages over objects containing only hatted quantities [as per Eqs.
(169)]. We therefore set these to zero, appealing to Sec. 4.3.4. For the same
reasons as in Sec. 5.4.2 this is not entirely legit at this point. But given
the complexity of the whole procedure I think it is justified for pedagogical
reasons to take the ‘illegal’ but shorter route first, in particular given
that the end result is correct. We will give a proper argument in Sec. 8.

Setting $\underline{P}$ and $\underline{\underline{L}}$ to zero, together with
Eqs. (164), allows us to make the following replacements in Eqs. (170) and
(171):

$\displaystyle i\widehat{M}(t)$ $\displaystyle\to$ $\displaystyle i\mu
P(t)=0,$ $\displaystyle i\widehat{P}(t)$ $\displaystyle\to$ $\displaystyle
i\mu M(t),$ $\displaystyle i\widehat{C}(t,t^{\prime})$ $\displaystyle\to$
$\displaystyle\frac{1}{2}\sigma^{2}L(t,t^{\prime})=0,$ $\displaystyle
i\widehat{K}(t,t^{\prime})$ $\displaystyle\to$
$\displaystyle\Gamma\sigma^{2}K(t^{\prime},t),$ $\displaystyle
i\widehat{L}(t,t^{\prime})$ $\displaystyle\to$
$\displaystyle\frac{1}{2}\sigma^{2}C(t,t^{\prime}).$ (172)

Additionally we introduce
$\underline{\underline{G}}\equiv-i\underline{\underline{K}}$, i.e., we make
the replacement $\underline{\underline{K}}\to i\underline{\underline{G}}$
(this is simply a change of notation).

We then have from Eq. (170) (recalling that
$\left\langle{\cdots}\right\rangle_{i}$ no longer depends on $i$ and has been
replaced by $\left\langle{\cdots}\right\rangle_{\star}$),

$\displaystyle\left\langle{g[\underline{x},\widehat{}\underline{x}]}\right\rangle_{\star}$
$\displaystyle=$
$\displaystyle\frac{1}{Z_{1}[\underline{\psi},\underline{h}]}\times\int{\cal
D}\underline{x}{\cal
D}\widehat{}\underline{x}\,p_{0}[x(0)]~{}g[\underline{x},\widehat{}\underline{x}]\exp\left(i\int
dt~{}\psi(t)x(t)\right)$ (173) $\displaystyle\times\exp\left(i\int
dt~{}\widehat{x}(t)\left(\frac{\dot{x}(t)}{x(t)}-[1-x(t)]-h(t)\right)\right)$
$\displaystyle\times\exp\left(-\frac{1}{2}\sigma^{2}\int
dt~{}dt^{\prime}\left[C(t,t^{\prime})\widehat{x}(t)\widehat{x}(t^{\prime})+2i\Gamma
G(t^{\prime},t)x(t)\widehat{x}(t^{\prime})\right]\right)$
$\displaystyle\times\exp\left(-i\mu\int dt~{}M(t)\widehat{x}(t)\right),$

and

$\displaystyle Z_{1}[\underline{\psi},\underline{h}]$ $\displaystyle=$
$\displaystyle\int{\cal D}\underline{x}{\cal
D}\widehat{}\underline{x}\,p_{0}[x(0)]~{}\exp\left(i\int
dt~{}\psi(t)x(t)\right)$ (174) $\displaystyle\times\exp\left(i\int
dt~{}\widehat{x}(t)\left(\frac{\dot{x}(t)}{x(t)}-[1-x(t)]-h(t)\right)\right)$
$\displaystyle\times\exp\left(-\frac{1}{2}\sigma^{2}\int
dt~{}dt^{\prime}\left[C(t,t^{\prime})\widehat{x}(t)\widehat{x}(t^{\prime})+2i\Gamma
G(t^{\prime},t)x(t)\widehat{x}(t^{\prime})\right]\right)$
$\displaystyle\times\exp\left(-i\mu\int dt~{}M(t)\widehat{x}(t)\right).$

###  7.2 Interpretation in terms of an effective single-species process

We now make the following statement:

For given $\underline{\underline{C}},\underline{\underline{G}}$ and
$\underline{M}$, the expression in Eq. (174) is the generating functional of
the effective dynamics

$\displaystyle\dot{x}(t)=x(t)\left[1-x(t)+\Gamma\sigma^{2}\int
dt^{\prime}G(t,t^{\prime})x(t^{\prime})+\mu M(t)+\eta(t)+h(t)\right],$ (175)

where $\eta(t)$ is Gaussian noise of zero average ($\mathbb{E}[\eta(t)]=0$ for
all $t$) and with correlations in time given by

$\mathbb{E}[\eta(t)\eta(t^{\prime})]=\sigma^{2}C(t,t^{\prime}).$ (176)

For the purpose of this statement (and its derivation below) we use the symbol
$\mathbb{E}$ for averages over realisations of $\underline{\eta}$.

Proof of the statement:
We set up the generating functional for the expression in Eq. (175), after
dividing through by $x(t)$ (I remind the reader that Appendix A contains a
brief description of a generating-functional calculation of the Lotka–Volterra
systems without this division). After enforcing the dynamics via suitable
delta functions in their exponential representation, we have

$\displaystyle Z[\underline{\psi}]$ $\displaystyle=$ $\displaystyle\int{\cal
D}\underline{x}{\cal D}\widehat{}\underline{x}\,p_{0}[x(0)]\,\exp\left(i\int
dt\,\psi(t)x(t)\right)$ $\displaystyle\times\exp\left(i\int
dt\,\widehat{x}(t)\left[\frac{\dot{x}(t)}{x(t)}-1+x(t)-\Gamma\sigma^{2}\int
dt^{\prime}G(t,t^{\prime})x(t^{\prime})-\mu M(t)-\eta(t)-h(t)\right]\right).$

This is the generating functional for the effective process, given a
realisation of $\underline{\eta}$. Therefore, we next need to average over
realisations of $\underline{\eta}$. The relevant term in Eq.
(LABEL:eq:gf_eff_vanilla) is $\exp\left(-i\int
dt\widehat{x}(t)\eta(t)\right)$. Using
$\mathbb{E}[\eta(t)\eta(t^{\prime})]=\sigma^{2}C(t,t^{\prime})$ and the rules
of Gaussian integration (Sec. 3.4), we have

$\displaystyle\mathbb{E}\left[\exp\left(i\int
dt\widehat{x}(t)\eta(t)\right)\right]$ $\displaystyle=$
$\displaystyle\exp\left(-\frac{1}{2}\sigma^{2}\int
dt\,dt^{\prime}C(t,t^{\prime})\widehat{x}(t)\widehat{x}(t^{\prime})\right).$
(178)

Thus, we find [with $Z[\underline{\psi}]$ as in Eq.
(LABEL:eq:gf_eff_vanilla)],

$\displaystyle\mathbb{E}\left[Z[\underline{\psi}]\right]$ $\displaystyle=$
$\displaystyle\int{\cal D}\underline{x}{\cal
D}\widehat{}\underline{x}\,p_{0}[x(0)]\,\exp\left(i\int
dt\,\psi(t)x(t)\right)$ (179) $\displaystyle\times\exp\left(i\int
dt\widehat{x}(t)\left[\frac{\dot{x}(t)}{x(t)}-1+x(t)-\Gamma\sigma^{2}\int
dt^{\prime}G(t,t^{\prime})x(t^{\prime})-\mu M(t)-h(t)\right]\right)$
$\displaystyle\times\exp\left(-\frac{1}{2}\sigma^{2}\int
dt\,dt^{\prime}C(t,t^{\prime})\widehat{x}(t)\widehat{x}(t^{\prime})\right).$

This, in turn, is exactly the generating functional in Eq. (174), which is
what we wanted to show.

∎

The statement we have just proved means that the average $\mathbb{E}$ (over
realisations of $\underline{\eta}$, i.e., realisations of the process in Eq.
(175)] is the same as the average $\left\langle{\dots}\right\rangle_{\star}$
in Eq. (173) [with $\underline{\psi}$ set to zero in Eq. (173)].