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Dynamical Large Deviations for Homogeneous Systems with Long Range Interactions and the Balescu–Guernsey–Lenard Equation

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Abstract

We establish a large deviation principle for time dependent trajectories (paths) of the empirical density of N particles with long range interactions, for homogeneous systems. This result extends the classical kinetic theory that leads to the Balescu–Guernsey–Lenard kinetic equation, by the explicit computation of the probability of typical and large fluctuations. The large deviation principle for the paths of the empirical density is obtained through explicit computations of a large deviation Hamiltonian. This Hamiltonian encodes all the cumulants for the fluctuations of the empirical density, after time averaging of the fast fluctuations. It satisfies a time reversal symmetry, related to the detailed balance for the stochastic process of the empirical density. This explains in a very simple way the increase of the macrostate entropy for the most probable states, while the stochastic process is time reversible, and describes the complete stochastic process at the level of large deviations.

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Acknowledgements

We warmly thank Oleg Zaboronski for first teaching us the Szegö–Widom theorem, used in this paper. We thank Julien Barré, Charles-Edouard Bréhier, Jules Guioth, and Julien Reygner for useful comments on our manuscript. The research leading to this work was supported by a sub-agreement from the Johns Hopkins University with funds provided by Grant No. 663054 from Simons Foundation. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of Simons Foundation or the Johns Hopkins University. We thank the two anonymous reviewers for their useful recommendations, which helped us to improve our first version of our manuscript.

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Appendices

Long Time Large Deviations for Quadratic Observables of Gaussian Processes, Functional Determinants and the Szegö–Widom Theorem for Fredholm Determinants

In this appendix, we explain how we can use the Szegö–Widom theorem in order to evaluate the large time asymptotics of Fredholm determinants that appears when computing the cumulant generating function of a quadratic observable of a Gaussian process. We follow the ideas in [14], adapting the discussion for the case of Gaussian processes with complex variables.

Let \(Y_{t}\) be a stationary \({\mathbb {C}}^{n}\)-valued Gaussian process with correlation matrix \(C\left( t\right) ={\mathbb {E}}\left( Y_{t}\otimes Y_{0}^{*}\right) \) and with a zero relation matrix \(R(t)={\mathbb {E}}\left( Y_{t}\otimes Y_{0}\right) =0\), let \(M\in {\mathscr {M}}_{n}\left( {\mathbb {C}}\right) \) be a \(n\times n\) Hermitian matrix. The aim of this appendix is to prove that

(58)

where \({\tilde{C}}\left( \omega \right) =\int _{{\mathbb {R}}}\text {e}^{i\omega t}C\left( t\right) \text {d}t\) is the Fourier transform of the correlation matrix \(C\left( t\right) \) and \(I_{n}\) is the \(n\times n\) identity matrix. We note that the determinant of the r.h.s. of (58) is a real number. Indeed, as \(Y_{t}\) is a stationary process, \({\tilde{C}}\left( \omega \right) \) and M are Hermitian matrices, then the determinant is the determinant of a Hermitian operator and is a real number.

For pedagogical reasons, in this appendix the result (58) is stated for a process \(Y_{t}\) that takes values in a finite-dimensional space. However with adapted hypotheses, this result can be generalized when \(Y_{t}\) is a stationary \({\mathscr {H}}\)-valued Gaussian process, where \({\mathscr {H}}\) is a Hilbert space, and where M is a Hermitian operator on \({\mathscr {H}}\).

In Sect. A.1, we state the Szegö–Widom theorem. In Sect. A.2, we explain that the left hand side of (58) is the logarithm of the determinant of a Gaussian integral, that this quantity can be expressed as a functional determinant for linear operators on \(L^{2}\left( \left[ 0,T\right] ,{\mathbb {C}}^{n}\right) \), and that the Szegoö–Widom theorem reduces it to the computation of frequency integrals of determinants of operators on the space \({\mathbb {C}}^{n}\), as expressed by (58).

1.1 The Szegö–Widom Theorem

We first define integral operators on \(L^{2}\left( \left[ 0,T\right] ,{\mathbb {C}}^{n}\right) \). We considers maps \(\varphi :\left[ 0,T\right] \rightarrow {\mathbb {C}}^{n}\) and \(K:{\mathbb {R}}\rightarrow {\mathscr {M}}_{n}\left( {\mathbb {C}}\right) \), where \({\mathscr {M}}_{n}\left( {\mathbb {C}}\right) \) is the set of \(n\times n\) complex matrices. We define the integral operator \({\mathbf {K}}_T\) by

$$\begin{aligned} {\mathbf {K}}_T\varphi \left( t\right) =\int _{0}^{T}K\left( t-s\right) \varphi \left( s\right) \text {d}s, \end{aligned}$$
(59)

\({\mathbf {K}}_T\) is a linear operator of \(L^{2}\left( \left[ 0,T\right] ,{\mathbb {C}}^{n}\right) \). K is called the kernel of the operator \({\mathbf {K}}_T\).

The Szegö–Widom theorem allows to compute large T asymptotics of the logarithm of the Fredholm determinant of the integral operator \(\text {Id}+{\mathbf {K}}_T\). The result is

$$\begin{aligned} \log \det _{\left[ 0,T\right] }\left( \text {Id}+{\mathbf {K}}_T\right) \underset{T\rightarrow \infty }{\sim }\frac{T}{2\pi }\int \text {d}\omega \,\log \det \left( I_{n}+\int _{{\mathbb {R}}}\text {e}^{i\omega t}K\left( t\right) \text {d}t\right) , \end{aligned}$$
(60)

where \(I_{n}\) is the \(n\times n\) identity matrix. Whereas the determinant on the l.h.s. of this expression, denoted by the subscript \(\left[ 0,T\right] \) is a Fredholm determinant, the determinant on the r.h.s. is a matrix determinant which can be more easily computed. Further details about this theorem and its possible applications can be found in [14].

1.2 Expectation of Functionals of Gaussian Processes

Let \(Y_{t}\) be a \({\mathbb {C}}^{n}\)-valued stationary Gaussian process with correlation matrix

$$\begin{aligned} C\left( t\right) ={\mathbb {E}}\left( Y_{t}\otimes Y_{0}^{*}\right) , \end{aligned}$$

and with zero relation matrix

$$\begin{aligned} R(t)={\mathbb {E}}\left( Y_{t}\otimes Y_{0}\right) =0. \end{aligned}$$

We will compute the large time asymptotics of

$$\begin{aligned} {\mathscr {U}}\left( T\right) =\log {\mathbb {E}}\exp \left( \int _{0}^{T}\text {d}t\,Y_{t}^{*\intercal }{\mathbf {M}}_T Y_{t}\right) , \end{aligned}$$

where \({\mathbf {M}}_T\) is an integral operator on \(L^{2}\left( \left[ 0,T\right] ,{\mathbb {C}}^{n}\right) \) whose integral kernel is given by \(M\left( t\right) \) (see the definition (59)). We assume that for all times t, \(M\left( t\right) \) is a \(n\times n\) Hermitian matrix. As \(Y_{t}\) is a Gaussian process we can compute the expectation as a Gaussian integral. It is straightforward to check that

$$\begin{aligned} {\mathbb {E}}\exp \left( \int _{0}^{T}\text {d}t\,Y_{t}^{*\intercal }{\mathbf {M}}_T Y_{t}\right) =\det _{\left[ 0,T\right] }\left( \text {Id}-({\mathbf {M}} {\mathbf {C}})_T \right) ^{-1}, \end{aligned}$$

where \((\mathbf {MC})_T\) is the integral operator whose kernel is \((M\star C)(t)\) the convolution product on [0, T] of the kernels M(t) and C(t).

Then, we can deduce the following expression for \(\mathscr {U}\)

$$\begin{aligned} {\mathscr {U}}\left( T\right) =-\log \det _{\left[ 0,T\right] }\left( \text {Id}-({\mathbf {M}}{\mathbf {C}})_T \right) , \end{aligned}$$

where the determinant is the Fredholm determinant of the integral operator \(\text {Id}-({\mathbf {M}}{\mathbf {C}})_T\). Generally, it is not obvious how to compute this kind of Fredholm determinant. Fortunately, we can use the Szegö–Widom theorem to obtain an expression for large T asymptotics as a finite-dimensional determinant. Using the result (60) from Sect. A.1, we get

$$\begin{aligned} {\mathscr {U}}\left( T\right) \underset{T\rightarrow \infty }{\sim }-\frac{T}{2\pi }\int \text {d}\omega \,\log \det \left( I_{n}-\int _{{\mathbb {R}}}\text {e}^{-i\omega t}\left( M\star C\right) \left( t\right) \text {d}t\right) . \end{aligned}$$

In the special case when \({\mathbf {M}}_T\) is a diagonal integral operator, i.e. when its kernel is \(M(t)=M\delta (t)\), we can write

$$\begin{aligned} {\mathscr {U}}\left( T\right) \underset{T\rightarrow \infty }{\sim }-\frac{T}{2\pi }\int \text {d}\omega \,\log \det \left( I_{n}-M\int _{{\mathbb {R}}}\text {e}^{-i\omega t}C\left( t\right) \text {d}t\right) , \end{aligned}$$

which is the result (58). In these expressions, the determinant to be computed on the r.h.s. is the determinant of a \(n\times n\) matrix.

Computation of the Determinant of the Operator \(u_{{\mathbf {k}},\omega }\)

In this appendix, we compute the determinant of the operator \(u_{{\mathbf {k}},\omega }\), encountered in Sect. 4.2, and defined by

$$\begin{aligned} u_{{\mathbf {k}},\omega }\left[ \varphi \right] \left( {\mathbf {v}}_{1}\right)= & {} \varphi \left( {\mathbf {v}}_{1}\right) -\int \text {d}{\mathbf {v}}_{2}\text {d}{\mathbf {v}}_{3}\,M\left( {\mathbf {k}},{\mathbf {v}}_{1},{\mathbf {v}}_{2}\right) \widetilde{{\mathscr {C}}_{GG}}\left( {\mathbf {k}},\omega ,{\mathbf {v}}_{2},{\mathbf {v}}_{3}\right) \varphi \left( {\mathbf {v}}_{3}\right) , \end{aligned}$$

for any \(\varphi \in {\mathscr {H}}_{{\mathbf {v}}}\), \({\mathscr {H}}_{{\mathbf {v}}}\) being the Hilbert space of complex functions over the velocity space. Using Eq. (33), we can simplify this expression

$$\begin{aligned} u_{{\mathbf {k}},\omega }\left[ \varphi \right] \left( {\mathbf {v}}_{1}\right) =\varphi \left( {\mathbf {v}}_{1}\right) -i{\hat{W}}\left( {\mathbf {k}}\right) {\mathbf {k}}\cdot \int \text {d}{\mathbf {v}}_{2}\text {d}{\mathbf {v}}_{3}\,\widetilde{{\mathscr {C}}_{GG}}\left( {\mathbf {k}},\omega ,{\mathbf {v}}_{2},{\mathbf {v}}_{3}\right) \left\{ \frac{\partial p}{\partial {\mathbf {v}}}\left( {\mathbf {v}}_{2}\right) -\frac{\partial p}{\partial {\mathbf {v}}}\left( {\mathbf {v}}_{1}\right) \right\} \varphi \left( {\mathbf {v}}_{3}\right) .\nonumber \\ \end{aligned}$$
(61)

We note that the operator \(u_{{\mathbf {k}},\omega }\) has the form

$$\begin{aligned} u_{{\mathbf {k}},\omega }:\varphi \longmapsto \varphi -\left\langle w,{\mathbf {Q}}\varphi \right\rangle v-\left\langle v,{\mathbf {Q}}\varphi \right\rangle w, \end{aligned}$$
(62)

where \({\mathbf {Q}}\) is a Hermitian operator over \(\mathscr {H_{{\mathbf {v}}}}\), w and v are complex functions over the velocity space, and \(\left\langle .,.\right\rangle \) denotes the Hermitian product: \(\left\langle a,b\right\rangle =\int \text {d}{\mathbf {v}}\,a^{*}\left( {\mathbf {v}}\right) b\left( {\mathbf {v}}\right) \). The connection is made between formulas (61) and (62) by setting \(v\left( {\mathbf {v}}\right) =-i{\mathbf {k}}.\frac{\partial p}{\partial {\mathbf {v}}}\) , \(w\left( {\mathbf {v}}\right) ={\hat{W}}\left( {\mathbf {k}}\right) \) and \({\mathbf {Q}}\left[ \phi \right] \left( {\mathbf {v}}_{1}\right) =\int \text {d}{\mathbf {v}}_{2}\,\widetilde{{\mathscr {C}}_{GG}}\left( {\mathbf {k}},\omega ,{\mathbf {v}}_{1},{\mathbf {v}}_{2}\right) \phi \left( {\mathbf {v}}_{2}\right) \). Using (28), we see that Q is a Hermitian operator. We note that, whenever \(\frac{\partial p}{\partial {\mathbf {v}}}\) is not a constant in the velocity space, v and w are linearly independent.

Formula (62) shows that \(u_{{\mathbf {k}},\omega }-\text {Id}\) is a rank two linear operator. Then \(\det _{{\mathscr {H}}_{{\mathbf {v}}}}u_{{\mathbf {k}},\omega }\) is the determinant of the operator \(u_{{\mathbf {k}},\omega }\) restricted to \(\text {span}{{\left( v,w\right) }}\):

$$\begin{aligned} \det _{{\mathscr {H}}_{{\mathbf {v}}}}u_{{\mathbf {k}},\omega }=\begin{vmatrix}1-\left\langle w,{\mathbf {Q}}v\right\rangle&-\left\langle w,{\mathbf {Q}}w\right\rangle \\ -\left\langle v,{\mathbf {Q}}v\right\rangle&1-\left\langle v,{\mathbf {Q}}w\right\rangle \end{vmatrix}. \end{aligned}$$

Then

$$\begin{aligned} \det _{{\mathscr {H}}_{{\mathbf {v}}}}u_{{\mathbf {k}},\omega }=1-2\mathfrak {R}\left[ \left\langle v,{\mathbf {Q}}w\right\rangle \right] +\left\langle v,{\mathbf {Q}}w\right\rangle \left\langle v,{\mathbf {Q}}w\right\rangle ^{*}-\left\langle w,{\mathbf {Q}}w\right\rangle \left\langle v,{\mathbf {Q}}v\right\rangle . \end{aligned}$$

where we have used \(\left\langle w,{\mathbf {Q}}v\right\rangle =\left\langle {\mathbf {Q}}w,v\right\rangle =\left\langle v,{\mathbf {Q}}w\right\rangle ^{*}\), as \({\mathbf {Q}}\) is an Hermitian operator.

We can explicitly compute the determinant of (61). We have

$$\begin{aligned} \left\langle v,{\mathbf {Q}}v\right\rangle =\int \text {d}{\mathbf {v}}_{1}\text {d}{\mathbf {v}}_{2}\,{\mathbf {k}}.\frac{\partial p}{\partial {\mathbf {v}}_{1}}{\mathbf {k}}.\frac{\partial p}{\partial {\mathbf {v}}_{2}}\widetilde{{\mathscr {C}}_{GG}}\left( {\mathbf {k}},\omega ,{\mathbf {v}}_{1},{\mathbf {v}}_{2}\right) , \\ \left\langle v,{\mathbf {Q}}w\right\rangle =i\int \text {d}{\mathbf {v}}_{1}\,{\mathbf {k}}.\frac{\partial p}{\partial {\mathbf {v}}_{1}}\widetilde{{\mathscr {C}}_{VG}}\left( {\mathbf {k}},\omega ,{\mathbf {v}}_{1}\right) ^{*}, \end{aligned}$$

and

$$\begin{aligned} \left\langle w,{\mathbf {Q}}w\right\rangle =\widetilde{{\mathscr {C}}_{VV}}\left( {\mathbf {k}},\omega \right) , \end{aligned}$$

where \(\widetilde{{\mathscr {C}}_{VG}}\), \(\widetilde{{\mathscr {C}}_{VV}}\) and \(\widetilde{{\mathscr {C}}_{GG}}\) are the two-point correlations functions of the quasi-linear problem computed in Sect. 3.2, and we have used (29-30).

We conclude that

$$\begin{aligned}&\underset{{\mathscr {H}}_{{\mathbf {v}}}}{\det }\left( u_{{\mathbf {k}},\omega }\right) =1+2\int \text {d}{\mathbf {v}}_{1}\,{\mathbf {k}}.\frac{\partial p}{\partial {\mathbf {v}}_{1}}\mathfrak {I}\left( \widetilde{{\mathscr {C}}_{VG}}\left( {\mathbf {k}},\omega ,{\mathbf {v}}_{1}\right) \right) \\&\quad +\int \text {d}{\mathbf {v}}_{1}\text {d}{\mathbf {v}}_{2}\,{\mathbf {k}}.\frac{\partial p}{\partial {\mathbf {v}}_{1}}{\mathbf {k}}.\frac{\partial p}{\partial {\mathbf {v}}_{2}}\left\{ \widetilde{{\mathscr {C}}_{VG}}\left( {\mathbf {k}},\omega ,{\mathbf {v}}_{1}\right) \widetilde{{\mathscr {C}}_{VG}}\left( {\mathbf {k}},\omega ,{\mathbf {v}}_{2}\right) ^{*}-\widetilde{{\mathscr {C}}_{VV}}\left( {\mathbf {k}},\omega \right) \widetilde{{\mathscr {C}}_{GG}}\left( {\mathbf {k}},\omega ,{\mathbf {v}}_{1},{\mathbf {v}}_{2}\right) \right\} . \end{aligned}$$

Consistency and Validation of the Cumulant Series Expansion

In this appendix, we expand H from the formula (47) in powers of p. This amounts at a cumulant expansion for the statistics of the fluctuations. We use this expansion to prove a conjecture made in the paper [23], and to recover the Gaussian (order two) truncation we computed in [23].

We expand the logarithm in formula (47) to obtain

$$\begin{aligned} H\left[ f,p\right] =\frac{1}{4\pi L^{3}}\sum _{{\mathbf {k}}}\int \text {d}\omega \,\sum _{n=1}^{+\infty }\frac{1}{n}\left( {\mathscr {J}}\left[ f,p\right] \left( {\mathbf {k}},\omega \right) \right) ^{n}=\sum _{n=1}^{+\infty }H^{(n)}\left[ f,p\right] . \end{aligned}$$
(63)

The second equality defines \(H^{(n)}\) as being the terms homogeneous of order n in p in this expansion. It is the n-th cumulant.

We also define \({\mathbf {B}}^{(m)}\) as

$$\begin{aligned} {\mathbf {B}}^{(m)}\left( {\mathbf {v}}_{1},\ldots ,{\mathbf {v}}_{2m}\right) =\frac{\left( 2\pi \right) ^{2m}}{4\pi mL^{3}}\sum _{{\mathbf {k}}}\int _{\Gamma }\text {d}\omega \,\frac{W\left( {\mathbf {k}}\right) ^{2m}}{\left| \varepsilon \left( {\mathbf {k}},\omega \right) \right| ^{2m}}{\mathbf {k}}^{\otimes 2m}\prod _{i=1}^{2m}\delta \left( \omega -{\mathbf {k}}.{\mathbf {v}}_{i}\right) . \end{aligned}$$

\({\mathbf {B}}^{(m)}\) is a rank 2m tensor. \(l^{(k)}\) and \(q^{(k)}\) are defined by the relations. We have \({\mathscr {J}}\left[ f,p\right] ={\mathscr {L}}\left[ f,p\right] +Q\left[ f,p,p\right] ,\) where \({\mathscr {L}}\) and Q are defined in Eqs. (44) and (45). We will need to compute \(\left( {\mathscr {L}}\left[ f,p\right] \right) ^{k}\), which is \({\mathscr {L}}\left[ f,p\right] \) to the power k. We define \(l^{(k)}\) and \(q^{(k)}\) by

$$\begin{aligned} \left( {\mathscr {L}}\left[ f,p\right] \right) ^{k}=\int \text {d}{\mathbf {v}}_{1}\cdots \text {d}{\mathbf {v}}_{2k}\,l^{(k)}\left[ f,p\right] \prod _{j=1}^{k}{\mathbf {A}}\left( {\mathbf {k}},\omega ,{\mathbf {v}}_{2j-1},{\mathbf {v}}_{2j}\right) , \end{aligned}$$

and

$$\begin{aligned} \left( Q\left[ f,p,p\right] \right) ^{k}=\int \text {d}{\mathbf {v}}_{1}\cdots \text {d}{\mathbf {v}}_{2k}\,q^{(k)}\left[ f,p,p\right] \prod _{j=1}^{k}{\mathbf {A}}\left( {\mathbf {k}},\omega ,{\mathbf {v}}_{2j-1},{\mathbf {v}}_{2j}\right) . \end{aligned}$$

\(l^{(k)}\) and \(q^{(k)}\) are both tensors of order 2k. \(l^{(k)}\) depends on p as a homogeneous function of order k. \(q^{(k)}\) depends on p as a homogeneous function of order 2k.

In the expansion of \(\left( {\mathscr {J}}\left[ f,p\right] \right) ^{n}\) using \({\mathscr {J}}\left[ f,p\right] ={\mathscr {L}}\left[ f,p\right] +Q\left[ f,p,p\right] ,\) we see that for all \(m\in \left[ n/2,n\right] \cap {\mathbb {N}}\), \({\mathscr {L}}^{2m-n}\left[ f,p\right] Q^{n-m}\left[ f,p,p\right] \) is homogeneous of order n in p. Using this remark, from Eq. (63) we obtain

$$\begin{aligned}&H^{(n)}\left[ f,p\right] =\sum _{m\in \left[ n/2,n\right] \cap {\mathbb {N}}}\int \text {d}{\mathbf {v}}_{1}\cdots \text {d}{\mathbf {v}}_{2m}\,\left( \begin{array}{c} m\\ 2m-n \end{array}\right) \frac{1}{\left( 2\pi \right) ^{2m}}{\mathbf {B}}^{(m)}\nonumber \\&\quad \left( {\mathbf {v}}_{1},\ldots ,{\mathbf {v}}_{2m}\right) :l^{(2m-n)}\left[ f,p\right] q^{(n-m)}\left[ f,p,p\right] , \end{aligned}$$
(64)

where the symbol \(``:\hbox {''}\) means a contraction of a tensor of order 2m with another tensor of order 2m.

This result ensures that as soon as \(n>2,\) \(H^{(n)}\) only includes terms proportional to the tensors \({\mathbf {B}}^{(m)}\) with \(m\ge n/2\ge 2.\) In [23], we used a conjecture on \(H^{(n)}\) to justify the Gaussian truncation of the cumulant series expansion of (22). The conjecture was that only the two first cumulants \(H^{(1)}\) and \(H^{(2)}\) do involve the tensor \({\mathbf {B}}={\mathbf {B}}^{(1)}\), whereas all the other cumulants \(H^{(n)}\) for \(n>2\) only involve tensors \({\mathbf {B}}^{(m)}\) with \(m\ge 2\). Expansion thus (64) justifies this conjecture.

In [23], from a cumulant series expansion, we obtained that

$$\begin{aligned} H\left[ f,p\right] =H_{\text {quad}}\left[ f,p\right] +O\left( p^{2}\right) , \end{aligned}$$

where

$$\begin{aligned}&H_{\text {quad}}\left[ f,p\right] =\int \text {d}{\mathbf {v}}_{1}\text {d}{\mathbf {v}}_{2}\,{\mathbf {B}}\left( {\mathbf {v}}_{1},{\mathbf {v}}_{2}\right) :\frac{\partial p}{\partial {\mathbf {v}}_{1}}\left\{ \frac{\partial f}{\partial {\mathbf {v}}_{2}}f({\mathbf {v}}_{1})-f({\mathbf {v}}_{2})\frac{\partial f}{\partial {\mathbf {v}}_{1}}\right\} \nonumber \\&\qquad +\int \text {d}{\mathbf {v}}_{1}\text {d}{\mathbf {v}}_{2}\,{\mathbf {B}}\left( {\mathbf {v}}_{1},{\mathbf {v}}_{2}\right) :\left\{ \frac{\partial p}{\partial {\mathbf {v}}_{1}}\frac{\partial p}{\partial {\mathbf {v}}_{1}}-\frac{\partial p}{\partial {\mathbf {v}}_{1}}\frac{\partial p}{\partial {\mathbf {v}}_{2}}\right\} f({\mathbf {v}}_{1})f({\mathbf {v}}_{2})\nonumber \\&\qquad +\int \text {d}{{{\mathbf {v}}_{1}}}\text {d}{\mathbf {v}}_{2}\text {d}{{{\mathbf {v}}_{3}}}\text {d}{\mathbf {v}}_{4}\,{\mathbf {B}}^{(2)}\left( {\mathbf {v}}_{1},{\mathbf {v}}_{2},{\mathbf {v}}_{3},{\mathbf {v}}_{4}\right) :\frac{\partial p}{\partial {\mathbf {v}}_{1}}\frac{\partial p}{\partial {\mathbf {v}}_{2}}\left\{ f({\mathbf {v}}_{1})f({\mathbf {v}}_{2})\frac{\partial f}{\partial {\mathbf {v}}_{3}}\frac{\partial f}{\partial {\mathbf {v}}_{4}}\right. \nonumber \\&\qquad \left. -2f({\mathbf {v}}_{1})\frac{\partial f}{\partial {\mathbf {v}}_{2}}f({\mathbf {v}}_{3})\frac{\partial f}{\partial {\mathbf {v}}_{4}}+\frac{\partial f}{\partial {\mathbf {v}}_{1}}\frac{\partial f}{\partial {\mathbf {v}}_{2}}f({\mathbf {v}}_{3})f({\mathbf {v}}_{4})\right\} , \end{aligned}$$
(65)

and where \(O\left( p^{2}\right) \) designates terms that are of order more than two in the conjugate momentum p, and the symbol \(``:\hbox {''}\) means a contraction of two tensors of order 2 and 4, for the two first lines and the third line, respectively. We see that \(H^{(1)}+H^{(2)}=H_{\text {quad}}.\) We conclude that (47) is consistent with the quadratic approximation of the large deviation Hamiltonian we obtained in [23].

Current Formulation of the Large Deviation Principle

Because the particle number is conserved, it is clear that the dynamics of the empirical density has a conservative form \(\frac{\partial f_{N}}{\partial t}+\frac{\partial }{\partial {\mathbf {v}}}\cdot {\mathbf {j}}_{N}=0\). For the microscopic dynamics (before time averaging), this is a consequence of Eqs. (15) or (19) with

$$\begin{aligned} {\mathbf {j}}_{N}\left( {\mathbf {v}},t\right) =-\frac{1}{NL^{3}}\int \text {d}{\mathbf {r}}\,\left( \frac{\partial V\left[ \delta g_{N}\right] }{\partial {\mathbf {r}}}\delta g_{N}\right) . \end{aligned}$$

After time averaging, we could have obtained the path large deviations by studying the large deviations of the time averaged current. Alternatively, we can rephrase our large deviation principle as a large deviation principle for the current, through a change of variable. This is the subject of this appendix.

The conservative nature of the dynamics is visible because the large deviation Hamiltonian H (47) does depend on the conjugate momentum p only through its gradient \(\partial p/\partial {\mathbf {v}}\). We define \({\tilde{H}}\) as \({\tilde{H}}\left[ f,\partial p/\partial {\mathbf {v}}\right] =H\left[ f,p\right] .\) We start from the definition of the large deviation Lagrangian

$$\begin{aligned} L\left[ f,{\dot{f}}\right] =\text {Sup}_{p}\left\{ \int \text {d}{\mathbf {v}}\,{\dot{f}}p-H[f,p]\right\} . \end{aligned}$$

Writing \({\dot{f}}\) as the divergence of a current \({\dot{f}}+\partial /\partial {\mathbf {v}}\cdot {\mathbf {j}}=0\), we have

$$\begin{aligned} L\left[ f,{\dot{f}}\right] =\underset{\left\{ {\mathbf {j}}\mid {\dot{f}}+\frac{\partial }{\partial {\mathbf {v}}}\cdot {\mathbf {j}}=0\right\} }{\text {Sup}}\text {Sup}_{p}\left\{ -\int \text {d}{\mathbf {v}}\,p\frac{\partial }{\partial {\mathbf {v}}}\cdot {\mathbf {j}}-H[f,p]\right\} . \end{aligned}$$

Using \(H\left[ f,p\right] ={\tilde{H}}\left[ f,\partial p/\partial {\mathbf {v}}\right] \), and integrating by part, we have

$$\begin{aligned} L\left[ f,{\dot{f}}\right] =\underset{\left\{ {\mathbf {j}}\mid {\dot{f}}+\frac{\partial }{\partial {\mathbf {v}}}\cdot {\mathbf {j}}=0\right\} }{\text {Sup}}{\tilde{L}}\left[ f,{\mathbf {j}}\right] \end{aligned}$$

with

$$\begin{aligned} {\tilde{L}}\left[ f,{\mathbf {j}}\right] =\underset{{\mathbf {E}}}{\text {Sup}}\left\{ \int \text {d}{\mathbf {v}}\,{\mathbf {j}}\cdot {\mathbf {E}}-{\tilde{H}}[f,{\mathbf {E}}]\right\} . \end{aligned}$$

where \({\mathbf {E}}\) designates the conjugate quantity of the current \({\mathbf {j}}\).

We thus have the large deviation principle

$$\begin{aligned} {\mathbf {P}}\left( \left\{ f_{N}(\tau )\right\} _{0\le \tau \le T}=\left\{ f(\tau )\right\} _{0\le \tau \le T}\right) \underset{N\rightarrow \infty }{\asymp }\text {e}^{-NL^{3}\underset{\left\{ {\mathbf {j}}\mid {\dot{f}}+\frac{\partial }{\partial {\mathbf {v}}}\cdot {\mathbf {j}}=0\right\} }{\text {Sup}}\int _{0}^{T}\text {d}\tau \,{\tilde{L}}\left[ f,{\mathbf {j}}\right] }\text {e}^{-NI_{0}\left[ f\left( \tau =0\right) \right] }. \end{aligned}$$
(66)

We note that we can also write a large deviation principle for the joint probability of the empirical density and the time averaged current \({\mathbf {j}}_{N}\left( \tau \right) \)

$$\begin{aligned} {\mathbf {P}}\left( \left\{ f_{N}(\tau ),{\mathbf {j}}_{N}\left( \tau \right) \right\} _{0\le t\le T}=\left\{ f(\tau ),{\mathbf {j}}\left( \tau \right) \right\} _{0\le t\le T}\right) \underset{N\rightarrow \infty }{\asymp }\text {e}^{-N{\mathscr {A}}\left[ f,{\mathbf {j}}\right] }\text {e}^{-NI_{0}\left[ f\left( \tau =0\right) \right] }, \end{aligned}$$

with

$$\begin{aligned} {\mathscr {A}}\left[ f,{\mathbf {j}}\right] ={\left\{ \begin{array}{ll} \begin{array}{llll} L^{3}\int _{0}^{T}\text {d}\tau {\tilde{L}}\left[ f,{\mathbf {j}}\right] &{} \quad \text {if}\quad {{{\dot{f}}}+{\frac{\partial }{\partial {\mathbf {v}}}\cdot }{{\mathbf {j}}}=0,}\\ +\infty &{}\quad \text {otherwise}. \end{array}\end{array}\right. } \end{aligned}$$

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Feliachi, O., Bouchet, F. Dynamical Large Deviations for Homogeneous Systems with Long Range Interactions and the Balescu–Guernsey–Lenard Equation. J Stat Phys 186, 22 (2022). https://doi.org/10.1007/s10955-021-02854-7

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