Fluctuations in Stochastic Interacting Particle Systems

Abstract

We discuss fluctuations in stochastic lattice gas models from a microscopic and mesoscopic perspective by using techniques from algebra, in particular the use of symmetries and time-reversal. First we present a generic method to derive rigorously duality functions. As applications we obtain detailed information about density fluctuations in the symmetric simple exclusion process on any graph and about the microscopic structure and fluctuations of shocks in the one-dimensional asymmetric simple exclusion process. Then we use time reversal to prove a general current fluctuation theorem from which celebrated fluctuation relations such as the Jarzynski relation and the Gallavotti-Cohen symmetry arise as corollaries and which can be straightforwardly generalized to derive other fluctuation relations. Finally, going beyond rigorous results, we describe briefly how nonlinear fluctuating hydrodynamics yields the Fibonacci family of dynamical universality classes which has the diffusive and Kardar-Parisi-Zhang universality classes as its first two members.

A Some Linear and Multilinear Algebra

In order to obtain more information about stochastic lattice gas models it will turn out to be convenient to write the generator of the process as a matrix which is called intensity matrix. Many probabilistic operations and notions then have a natural counterpart in linear algebra which we summarize here in elementary form. Indeed, much of what is presented is trivial, but perhaps necessary to write down in order to point out technically important subtleties and to introduce notation for the less common applications such the Kronecker product of matrices, sometimes also called outer product, which is essential for the choice of basis of the intensity matrix for lattice models.

In the following and throughout this work we use the Kronecker-symbol defined by

$$\begin{aligned} \delta _{\alpha ,\beta } = \left\{ \begin{array}{ll} 1 &{} \mathrm{if }~ \alpha =\beta \\ 0 &{} \mathrm{else } \end{array} \right. \end{aligned}$$

(A.1)

for $\alpha ,\beta$ from any set. Complex conjugation is denoted by a bar as e.g. in $\overline{z}$.

Matrices and Vectors. A $m\times n$ matrix A is a number array

$$\begin{aligned} A = \left( \begin{array}{cccc} A_{11} &{} A_{12} &{} A_{13} &{} \dots \\ A_{21} &{} A_{22} &{} A_{23} &{} \dots \\ A_{31} &{} A_{32} &{} A_{33} &{} \dots \\ A_{41} &{} A_{42} &{} A_{43} &{} \dots \\ A_{51} &{} A_{52} &{} A_{53} &{} \dots \\ \vdots &{} \vdots &{} \vdots &{} \ddots \end{array} \right) . \end{aligned}$$

with $m \ge 1$ rows and $n\ge 1$ columns and matrix elements $A_{kl}$ in row k and column l. The matrix elements $A_{kl}$ will be mostly real numbers, but they can also be complex in certain applications. Hence we shall generally assume . We discuss some special cases.

(a) If $m=n=1$ the matrix reduces a single number and we shall not differentiate between numbers and $1\times 1$-matrices.

(b) If $n=1$ and $m>1$ a matrix $\varPhi$ is a column array of m numbers. We call such a matrix a ket-vector that we denote by the so-called ket-symbol $| \, {\varPhi }\, \rangle$. The matrix elements $\varPhi _{1l}$ with $1\le l \le m$ will be denoted in simplified form by $\varPhi _{l}$ and called components of the ket-vector. Thus

$$\begin{aligned} | \, {\varPhi }\, \rangle = \left( \begin{array}{c} \varPhi _1 \\ \varPhi _2 \\ \vdots \\ \varPhi _m \end{array} \right) . \end{aligned}$$

The vector with $\varPhi _i = \delta _{ik}$ is a canonical basis vector of the vector space denoted by $| \, {e_k}\, \rangle$. The set $B_m := \{| \, {e_k}\, \rangle : k\in \{1,\dots ,m\}\}$ spans and is called the canonical basis.

(c) If $m=1$ and $n>1$ a matrix $\varPsi$ is a row array of n numbers. We call such a matrix a bra-vector that we denote by the so-called bra-symbol $\langle \, {\varPsi }\, |$. The matrix elements $\varPsi _{k1}$ with $1\le k \le n$ will be denoted in simplified form as $\varPsi _{k}$ and called components of the bra-vector. Thus

$$\begin{aligned} \langle \, {\varPsi }\, | = \left( \varPsi _1 , \varPsi _2 , \dots , \varPsi _n \right) . \end{aligned}$$

Defining $\langle \, {e_k}\, |=| \, {e_k}\, \rangle ^T$ one realizes that the set $B^*:= \{\langle \, {e_k}\, |: k\in \{1,\dots ,n\}\}$ spans . Since any finite-dimensional vector space is isomorphic to its dual, we can think of the bra-vectors $\mathbb {B}_n^*$ as representing the canonical basis of the dual space .

The letter or number inside the ket-symbol $| \, {\cdot }\, \rangle$ or the bra-symbol $\langle \, {\cdot }\, |$ is not to be understood as the argument of some function, but just as a symbol that collectively represents the components of the vector. When we use the term matrix we shall tacitly assume that $m,n\ge 2$. The distinction between “proper” matrices on the one hand and the two types of vectors or simple numbers on the other hand is useful because many fundamental linear algebra operations can be represented as products involving numbers, bra- and ket-vectors and proper matrices with more than one column or row.

We usually denote proper matrices by capital letters or small letters with circumflex accent as e.g. in $\hat{a}$. The unit matrix of dimension $n>2$ with components $A_{kl}=\delta _{k,l}$ is denoted by $\mathbf {1}$ and for $n=2$ we use the notation $\mathbbm {1}$. Since multiplication of a vector with the unit matrix is the same as multiplication with the scalar unity 1 of the field F we do not usually differentiate between the two operations, i.e., in equations for matrices we often write a multiple $x \mathbf {1}$ of the unity matrix simply as x.

Addition and Multiplication of Matrices. Any two matrices A and B which have the same number of rows and columns can be multiplied by a number and added to form a matrix $C=xA+yB$ with the rule that $C_{kl}=xA_{kl}+yB_{kl}$ where . Square matrices with $m=n$ form a ring with a multiplication rule that can be generalized to non-square matrices as follows.

Definition 12

(Matrix product). For $m,n,p \ge 1$ let A be a $m\times p$-matrix and B be a $p\times n$-matrix, both with matrix elements in some field F. The matrix product AB is an $m\times n$ matrix C with matrix elements $C_{kl} \equiv (AB)_{kl} \in F$ given by

$$\begin{aligned} C_{kl} = \sum _{j=1}^p A_{kj} B_{jl}, \quad 1 \le k \le m, \quad 1 \le l \le n. \end{aligned}$$

(A.2)

Square matrices A, B of the same dimension $m=n=p$ satisfying

$$\begin{aligned}{}[{A}\,,\,{B}] := AB - BA = 0 \end{aligned}$$

(A.3)

are said to commute.

Notice that unless $m=n$ the reverse product BA is not defined since the number of columns in the first factor must be equal to the number of rows in the second factor of any matrix product. For a square matrix A the $p^{th}$ power of A is denoted $A^p$ and is defined for strictly positive integers by iteration of (A.2). By convention $A^0 = \mathbf {1}$.

We discuss separately the special cases where at least one of the three number m, n, p is equal to one.

(a) If $n=1$ and $p,m>1$ the we can write the matrix B as a ket-vector $| \, {\varPhi }\, \rangle$ with components $\varPhi _k := B_{k1}$, $k\in \{1,\dots ,p\}$. Then also the matrix product C is a ket-vector (with m components given by (A.2)) and the matrix product can be interpreted as a linear mapping $| \, {\varPhi }\, \rangle \mapsto | \, {\tilde{\varPhi }}\, \rangle$ given by $| \, {\tilde{\varPhi }}\, \rangle = A | \, {\varPhi }\, \rangle$, corresponding to the standard right multiplication of a matrix A with the column vector $| \, {\varPhi }\, \rangle$.

(b) Likewise, for $m=1$ and $p,n>1$ we can write $A = \langle \, {\varPsi }\, |$ as a bra-vector with p components with components $\varPsi _l := A_{1l}$ and find that the matrix product is a linear mapping $\langle \, {\varPsi }\, | \mapsto \langle \, {\tilde{\varPsi }}\, |$ that yields the bra-vector $\langle \, {\tilde{\varPhi }}\, | = \langle \, {\varPsi }\, | B$ with n components given by (A.2), corresponding to the left multiplication of a matrix B with the row vector $\langle \, {\varPsi }\, |$.

(c) If $p=1$ and $m,n>1$ then the matrix product actually turns into a product of two vectors. It maps a m-component ket-vector $| \, {\varPhi }\, \rangle$ (=$m\times 1$-matrix A) with components $\varPhi _k := A_{k1}$ and an n-component bra-vector $\langle \, {\varPsi }\, |$ (=$1\times n$-matrix B) with components $\varPsi _l := B_{1l}$ into a proper $m\times n$ matrix

$$\begin{aligned} C=| \, {\varPhi }\, \rangle \langle \, {\varPsi }\, | \end{aligned}$$

(A.4)

with matrix elements $C_{kl} = \varPhi _k \varPsi _l$ as given by (A.2). This mapping, called dyadic product, is a special form of the Kronecker product discussed below.

(d) For $m=n=1$ the matrix product reduces to a single number $C=\langle \, {\varPsi }\, || \, {\varPhi }\, \rangle =C_{11} \in F$ with

$$\begin{aligned} \langle \, {\varPsi }\, || \, {\varPhi }\, \rangle = \sum _{i=1}^p \varPsi _i \varPhi _i \equiv \langle {\varPsi }| \, {\varPhi } \rangle . \end{aligned}$$

(A.5)

It defines a bilinear mapping $(\langle \, {\varPsi }\, |,| \, {\varPhi }\, \rangle )\mapsto C_{11}$ which can be interpreted as a dual pairing $d: \mathfrak {V}^*\times \mathfrak {V} \rightarrow F$, $(\langle \, {\varPsi }\, |,| \, {\varPhi }\, \rangle ) \mapsto \langle {\varPsi }| \, {\varPhi } \rangle$ since it is natural to regard the bra-vector to be an element of the vector space dual to the vector space to which the ket-vector belongs. This motivates the simplified notation $\langle {\varPsi }| \, {\varPhi } \rangle$ of this matrix product with only one vertical bar.

Specifically, for the basis vectors we obtain from (A.5) the biorthogonality relation

$$\begin{aligned} \langle {e_i}| \, {e_j} \rangle = \delta _{ij}. \end{aligned}$$

(A.6)

Notice the difference between the dual pairing (A.5) and the scalar product $s: \mathfrak {V} \times \mathfrak {V} \rightarrow F$ defined by the sesquilinear form $(| \, {\varPhi '}\, \rangle ,| \, {\varPhi }\, \rangle )\mapsto \langle {\varPhi ',\varPhi } \rangle :=\sum _{i=1}^p \overline{\varPhi }_i' \varPhi _i$ which is linear in the second argument, but antilinear in the first. When $\langle \, {\varPhi '}\, |$ has only real components (as is the case in most of our applications) this distinction is irrelevant, but should nevertheless be kept in mind.

The Kronecker Product. The Kronecker product $A\otimes B$ is defined for arbitrary rectangular matrices (including vectors and numbers) as follows.

Definition 13

(Kronecker product). Let A and B be two finite-dimensional matrices with $m_A\ge 1$ ($m_B\ge 1$) rows and $n_A\ge 1$ ($n_B\ge 1$) columns with matrix elements $A_{ij}$ and $B_{kl}$ respectively. The Kronecker product $A\otimes B$ is a $m_Am_B \times n_An_B$-matrix C with matrix elements

$$\begin{aligned} C_{(i-1)m_B+k,(j-1)n_B+l}=A_{ij} B_{kl} \end{aligned}$$

(A.7)

with $i \in \{1,\dots ,m_A\}, j \in \{1,\dots ,n_A\}, k \in \{1,\dots ,m_B\}, l \in \{1,\dots ,n_B\}$.

Alternatively we can write

$$\begin{aligned} A\otimes B = \left( \begin{array}{cccc} A_{11} B &{} A_{12} B &{} A_{13} B &{} \dots \\ A_{21} B &{} A_{22} B &{} A_{23} B &{} \dots \\ A_{31} B &{} A_{32} B &{} A_{33} B &{} \dots \\ A_{41} B &{} A_{42} B &{} A_{43} B &{} \dots \\ A_{51} B &{} A_{52} B &{} A_{53} B &{} \dots \\ \vdots &{} \vdots &{} \vdots &{} \ddots \end{array} \right) . \end{aligned}$$

Here each matrix “element” is itself a matrix, viz. the matrix B multiplied by the number $A_{ij}$. In general $A\otimes B\ne B\otimes A$. For the p-fold Kronecker product of a matrix A with itself is denoted by $A^{\otimes p}$ with the convention that $A^{\otimes 1} := A$ and $A^{\otimes 0} := 1$ where 1 is the unit element of F and not the unit matrix. We discuss special cases.

(a) Consider $n_A=n_B=1$, i.e., the Kronecker product of ket-vectors $| \, {\varPhi ^1}\, \rangle , | \, {\varPhi ^2}\, \rangle$ with components $\varPhi ^1_i$ where $i \in \{1,\dots ,m_A\}$ and $\varPhi ^2_k$ where $k \in \{1,\dots ,m_B\}$. The tensor product $| \, {\varPhi ^1}\, \rangle \otimes | \, {\varPhi ^2}\, \rangle$ is a column vector of dimension $m_A m_B$ denoted by $| \, {\varPhi ^1,\varPhi ^2}\, \rangle$ and has factorized components $(| \, {\varPhi ^1,\varPhi ^2}\, \rangle )_{(i-1)m_B+k} = \varPhi ^1_i \varPhi ^2_k$. Specifically, for the canonical basis vectors one gets $| \, {e_i}\, \rangle \otimes | \, {e_k}\, \rangle \equiv | \, {e_i,e_k}\, \rangle = | \, {e_{(i-1)m_B+k}}\, \rangle$. Thus the Kronecker product of two canonical basis vectors yields a canonical basis vector. The set $\mathbb {B}_{m_Am_B} := \{| \, {e_{(i-1)m_B+k}}\, \rangle : (i,k) \in \{1,\dots ,m_A\}\times \{1,\dots ,m_B\}\}$ forms the canonical basis of the tensor space .

(b) Similarly, for $m_A=m_B=1$, i.e., for bra-vectors $\langle \, {\varPsi ^1}\, |, \langle \, {\varPsi ^2}\, |$ with components $\varPsi ^1_j$ where $j \in \{1,\dots ,n_A\}$ and $\varPsi ^2_l$ where $l \in \{1,\dots ,n_B\}$ the tensor product $\langle \, {\varPsi ^1}\, | \otimes \langle \, {\varPsi ^2}\, |$ is a row vector of dimension $n_A n_B$ denoted by $\langle \, {\varPsi ^1,\varPsi ^2}\, |$. It has factorized components $(\langle \, {\varPsi ^1,\varPsi ^2}\, |)_{(j-1)n_B+l} = \varPsi ^1_j \varPsi ^2_l$ and for the canonical basis vectors one gets $\langle \, {e_j,e_l}\, | = \langle \, {e_{(j-1)n_B+l}}\, |$.

(c) For the Kronecker product of a bra-vector $\langle \, {\varPsi }\, |$ and a ket-vector $| \, {\varPhi }\, \rangle$ the Definition 13 yields

$$\begin{aligned} \langle \, {\varPsi }\, |\otimes | \, {\varPhi }\, \rangle = | \, {\varPhi }\, \rangle \otimes \langle \, {\varPsi }\, | = | \, {\varPhi }\, \rangle \langle \, {\varPsi }\, | \end{aligned}$$

(A.8)

with the dyadic product (A.4).

The Kronecker product is associative. Multiple Kronecker products of matrices define multilinear maps of the multiple tensor product of vector spaces defined by iterating the Kronecker product Definition 13. They satisfy the multiplication rule

$$\begin{aligned} (A\otimes B) (C \otimes D) = (AC) \otimes (BD) \end{aligned}$$

(A.9)

where we assume that the matrix products AC and BD are defined by (A.2).

We note an important factorization property of the dual pairing of Kronecker products of vectors which is an immediate consequence of the multilinearity of the Kronecker product encoded in (A.7).

Proposition 5

Let $\langle \, {\varPsi ^k}\, |$ ($| \, {\varPhi ^k}\, \rangle$) be a bra-vector (ket-vector) of dimension $d_k$ with components () and $\langle \, {\varPsi ^1,\varPsi ^2,\dots ,\varPsi ^L}\, | = \langle \, {\varPsi ^1}\, | \otimes \langle \, {\varPsi ^2}\, | \otimes \dots \otimes \langle \, {\varPsi ^L}\, |$ ($| \, {\varPhi ^1,\varPhi ^2,\dots ,\varPhi ^L}\, \rangle = | \, {\varPhi ^1}\, \rangle \otimes | \, {\varPhi ^2}\, \rangle \otimes \dots \otimes | \, {\varPhi ^L}\, \rangle$) be the L-fold Kronecker product of these vectors. Then the dual pairing factorizes as

$$\begin{aligned} \langle {\varPsi ^1,\varPsi ^2,\dots ,\varPsi ^L}| \, {\varPhi ^1,\varPhi ^2,\dots ,\varPhi ^L} \rangle = \prod _{k=1}^L \langle {\varPsi ^k}| \, {\varPhi ^k} \rangle \end{aligned}$$

(A.10)

with $\langle {\varPsi ^k}| \, {\varPhi ^k} \rangle$ given by (A.5).

When $\langle \, {\varPsi ^k}\, | = \langle \, {\varPsi }\, |$ for all $k \in \{1,\dots ,L\}$ then we write $\langle \, {\varPsi ^1,\varPsi ^2,\dots ,\varPsi ^L}\, | = \langle \, {\varPsi }\, |^{\otimes L}$ and analogously for ket-vectors and proper matrices A.

Finally we introduce local operators which act non-trivially only on component k in an L-fold tensor space. For simplicity we assume equal dimensions $d:=d_1=d_2=\dots =d_L$.

Definition 14

(Local operator). Let $\mathbf {1}$ be the d-dimensional unit matrix and A be an arbitrary square matrix of dimension $d \ge 1$. The local operator $A_k$ is the Kronecker product

$$\begin{aligned} A_k := \mathbf {1}^{\otimes (k-1)} \otimes A \otimes \mathbf {1}^{\otimes (L-k)}. \end{aligned}$$

(A.11)

Notice the difference between the number and the unit matrix $\mathbf {1}$ in this definition. The expression “local operator” come from the fact that when acting on a tensor vector $| \, {\varPhi ^1,\dots ,\varPhi ^L}\, \rangle$ only the $k^{th}$ factor is changed by the action of $A_k$. More precisely,

$$\begin{aligned} A_k \left( | \, {\varPhi ^1}\, \rangle \otimes \dots \otimes | \, {\varPhi ^k}\, \rangle \otimes \dots \otimes | \, {\varPhi ^L}\, \rangle \right) = | \, {\varPhi ^1}\, \rangle \otimes \dots \otimes | \, {\tilde{\varPhi }^k}\, \rangle \otimes \dots \otimes | \, {\varPhi ^L}\, \rangle . \end{aligned}$$

(A.12)

where $| \, {\tilde{\varPhi }^k}\, \rangle = A| \, {\varPhi ^k}\, \rangle$.

From (A.9) one finds

$$\begin{aligned} A_k B_k = (AB)_k \end{aligned}$$

(A.13)

which is equal to $B_kA_k$ if and only if $AB=BA$. On the other hand, by construction one has for two square matrices A, B of dimension k the commutation relation

$$\begin{aligned} A_k B_l = B_l A_k ~\mathrm{for}~ k\ne l \end{aligned}$$

(A.14)

even when $AB\ne BA$. In order to avoid confusion concerning the role of the indices we point out that for $L=2$ and $[{A}\,,\,{B}]\ne 0$ we have

$$\begin{aligned} A\otimes B = A_1 B_2 = B_2 A_1 \ne B \otimes A = B_1 A_2 = A_2 B_1. \end{aligned}$$

(A.15)

We also note that for matrices $A^{(k)}$ one has

$$\begin{aligned} A^{(1)}_1 A^{(2)}_2 \dots A^{(L)}_L = A^{(1)} \otimes A^{(2)} \otimes \dots \otimes A^{(L)}. \end{aligned}$$

(A.16)

The upper index defines the matrix while the lower index defines its position in the L-fold Kronecker product. We stress that $A^{(k)}$ is a matrix of dimension d while $A^{(k)}_k$ is a matrix of dimension $d^L$.

From Proposition 5 one finds for d-dimensional square matrices $A^{(k)}$ the factorization property

$$\begin{aligned} \langle \, {\varPsi ^1,\varPsi ^2,\dots ,\varPsi ^L}\, | A^{(1)}_1 A^{(2)}_2 \dots A^{(L)}_L | \, {\varPhi ^1,\varPhi ^2,\dots ,\varPhi ^L}\, \rangle = \prod _{k=1}^L \langle \, {\varPsi ^k}\, |A^{(k)}| \, {\varPhi ^k}\, \rangle . \end{aligned}$$

(A.17)

We write explicitly two special cases of particular importance:

$$\begin{aligned} \frac{\langle \, {\varPsi ^1,\varPsi ^2,\dots ,\varPsi ^L}\, | A_k | \, {\varPhi ^1,\varPhi ^2,\dots ,\varPhi ^L}\, \rangle }{\langle {\varPsi ^1,\varPsi ^2,\dots ,\varPsi ^L}| \, {\varPhi ^1,\varPhi ^2,\dots ,\varPhi ^L} \rangle }= & {} \frac{\langle \, {\varPsi ^k}\, |A| \, {\varPhi ^k}\, \rangle }{\langle {\varPsi ^k}| \, {\varPhi ^k} \rangle } \end{aligned}$$

(A.18)

$$\begin{aligned} \left( \langle \, {\varPsi }\, |\right) ^{\otimes L} \left( | \, {\varPhi }\, \rangle \right) ^{\otimes L}= & {} \langle {\varPsi }| \, {\varPhi } \rangle ^L. \end{aligned}$$

(A.19)

These computational properties of the matrix product (A.2) and of the Kronecker product defined in Definition 13 will be exploited throughout these notes.