Finite Size Corrections to the Large Deviation Function of the Density in the One Dimensional Symmetric Simple Exclusion Process

Abstract

The symmetric simple exclusion process is one of the simplest out-of-equilibrium systems for which the steady state is known. Its large deviation functional of the density has been computed in the past both by microscopic and macroscopic approaches. Here we obtain the leading finite size correction to this large deviation functional. The result is compared to the similar corrections for equilibrium systems.

Appendices

Appendix A: Additivity formulae

It is known [3, 30, 31] and has been used in several previous works [12, 13] that the steady state measure of the SSEP with injection and removal rates α,β,γ,δ, as defined in the introduction, can be calculated by the matrix ansatz [30, 31]. The probability of any microscopic configuration {n ₁,…,n _L } (with n _i =0 or 1) is given by

$$ P\bigl(\{n_1, \ldots, n_L\} \bigr) = \frac{\langle W | X_1 X_2 \ldots X_L | V \rangle}{ \langle W | (D+E)^L |V \rangle} $$

(55)

where each matrix X _i depends on the occupation n _i of site i

$$ X_i = n_i D + (1 - n_i) E $$

(56)

and the matrices D,E and the vectors |V〉,〈W| satisfy the following algebraic rules

$$\begin{aligned} & DE-ED= D+E \\ & \langle W | ( \alpha E - \gamma D) = \langle W| \\ & ( \beta D - \delta E)| V \rangle= | V \rangle . \end{aligned}$$

(57)

Given these algebraic rules, one can define a family of left and right eigenvectors 〈ρ _a ,a| and |ρ _b ,b〉 by

$$ \langle\rho_a,a| \bigl(\rho_a E - (1- \rho_a) D \bigr) = a \langle\rho_a,a| $$

(58)

$$ \bigl( (1-\rho_b) D - \rho_b E \bigr) | \rho_b,b \rangle= b |\rho_b,b \rangle . $$

(59)

The vectors 〈W| and |V〉 which appear in (55), (57) are examples of such eigenvectors

$$ \langle W| =\langle\rho_a,a| ,\qquad | V \rangle= | \rho_b,b \rangle $$

(60)

when ρ _a =α/(α+γ), ρ _b =δ/(δ+β) and a=1/(α+γ) , b=1/(δ+β) as in (1), (10).

Then for 0<b<1 and ρ _a >ρ _b , one can prove the following key additivity formula

$$ \frac{\langle\rho_a,a| X_1 X_2 |\rho_b,b \rangle}{ \langle\rho_a,a| \rho_b,b \rangle} = \oint\frac{d \rho }{ 2 \pi i}\frac{ (\rho_a - \rho_b)^{a+b}}{(\rho _a-\rho)^{a+b} (\rho-\rho_b) }\frac{\langle\rho_a,a| X_1 |\rho,b \rangle}{ \langle \rho_a,a|\rho,b \rangle} \frac{\langle\rho,1-b| X_2 |\rho_b,b \rangle}{\langle\rho,1-b|\rho_b,b \rangle} $$

(61)

where X ₁ and X ₂ are arbitrary polynomials of D’s and E’s and the contour is such that ρ _b <|ρ|<ρ _a .

Proof of (61)

Let us first derive of the following identity [13]

$$ \frac{\langle\rho_a,a| (D+E)^L |\rho_b,b \rangle }{\langle\rho_a,a |\rho_b,b \rangle} = \frac{\varGamma(a+b+ L)}{\varGamma(a+b) (\rho_a - \rho_b)^L} . $$

(62)

To do so one can notice that in the steady state of the SSEP, as defined in the introduction, the average occupations satisfy

$$\alpha- (\alpha+ \gamma) \langle n_1 \rangle= \langle n_1 - n_2 \rangle = \cdots\langle n_i - n_{i+1} \rangle = \cdots= ( \beta+ \delta) \langle n_L \rangle- \delta . $$

These L equations which express simply that in the steady state the current is conserved, can be solved. From the solution (14) one can see that

$$\langle n_i- n_{i+1} \rangle=\frac{(\rho_a- \rho_b) }{ L + a+b-1} . $$

On the other hand using the matrix representation (55), (57), (60) one has

$$\begin{aligned} \langle n_i - n_{i+1} \rangle =& \frac{\langle\rho_a,a|(D+E)^{i-1}(DE-ED) (D+E)^{L-i-1} |\rho_b,b \rangle}{ \langle\rho_a,a |(D+E)^L|\rho_b,b \rangle} \\ =& \frac{\langle\rho_a,a| (D+E)^{L-1} |\rho_b,b \rangle}{\langle\rho _a,a |(D+E)^L|\rho_b,b \rangle} . \end{aligned}$$

These two identities give the recursion

$$\frac{\langle\rho_a,a| (D+E)^{L-1} |\rho_b,b \rangle }{\langle\rho _a,a |(D+E)^L |\rho_b,b \rangle} = \frac{(\rho_a- \rho_b) }{ L + a+b-1} $$

which establishes the veracity of (62).

Now to prove (61) (as in [3]) one can first notice that the discussion can be limited to X ₁ and X ₂ of the form

$$\begin{aligned} X_1 &= \bigl[\rho_a E - (1-\rho_a) D \bigr]^{p_1} [ D + E]^{q_1} \\ X_2 &= [ D + E]^{p_2} \bigl[ (1-\rho_b) D - \rho_b E\bigr]^{q_2} \end{aligned}$$

as any polynomial in D’s and E’s can be written as a sum of such terms (this is because D and E are linear functions of the operators A and B defined by A=D+E and B=ρ _a E−(1−ρ _a )D and that AB−BA=A. Thus any word made up of A’s and B’s can be ordered as a sum of terms of the form $B^{p_{1}} A^{q_{1}}$ or $A^{p_{2}} B^{q_{2}}$). Then the left hand side of (61) becomes

$$ \frac{\langle\rho_a,a| X_1 X_2 |\rho_b,b \rangle}{ \langle\rho_a,a| \rho_b,b \rangle} = a^{p_1} b^{q_2} \frac{\langle\rho_a,a| (D+E)^{{q_1+p_2}} |\rho_b,b \rangle}{\langle \rho_a,a| \rho_b,b \rangle} $$

(63)

while the right hand side of (61) becomes

$$ a^{p_1} b^{q_2} \oint\frac{d \rho }{2 \pi i} \frac{(\rho_a - \rho _b)^{a+b} }{ (\rho_a-\rho)^{a+b} (\rho-\rho_b) }\frac{\langle\rho_a,a| (D+E)^{q_1} |\rho,b \rangle}{ \langle\rho_a,a|\rho,b \rangle} \frac{\langle\rho,1-b| (D+E)^{p_2} |\rho_b,b \rangle}{ \langle\rho,1-b|\rho_b,b \rangle} $$

(64)

and the equality of (63) and (64) follows from the expression (62) and the Cauchy theorem. This completes the derivation of (61).

First consequence of (61)

It is possible to show directly from the algebra (55), (57) that

$$\frac{\langle\rho_a,a| D |\rho_b,b \rangle}{\langle\rho_a,a| \rho_b,b \rangle} = \frac{ b \rho_a + a \rho_b}{ \rho_a - \rho_b} ; \qquad \frac{\langle\rho_a,a| E |\rho_b,b \rangle}{ \langle\rho_a,a| \rho_b,b \rangle} = \frac{b(1-\rho_a) + a ( 1-\rho_b) }{\rho_a - \rho_b} $$

which becomes by replacing ρ _a by ρ and a by 1−b

$$\frac{\langle\rho,1-b| D |\rho_b,b \rangle}{\langle\rho,1-b| \rho_b,b \rangle} = b + \frac{ \rho_b }{ \rho- \rho_b} ; \qquad \frac{\langle\rho,1-b| E |\rho_b,b \rangle}{ \langle\rho,1-b| \rho_b,b \rangle} = - b + \frac{ 1-\rho_b }{ \rho- \rho_b} . $$

Therefore (61) becomes after integration

$$\begin{aligned} \frac{\langle\rho_a,a| X_0 D |\rho_b,b \rangle}{ \langle\rho_a,a| \rho_b,b \rangle} &= b \frac{\langle\rho_a,a| X_0 |\rho_b,b \rangle}{ \langle\rho_a,a|\rho _b,b \rangle} \\ &\quad+ \rho_b \frac{ d }{ d \rho} \biggl[ \biggl( \frac{\rho_a - \rho_b }{ \rho_a - \rho} \biggr)^{a+b} \frac{\langle\rho_a,a| X_0 |\rho,b \rangle}{ \langle\rho_a,a|\rho,b \rangle} \biggr] \bigg\vert _{\rho=\rho_b} , \end{aligned}$$

(65)

$$\begin{aligned} \frac{\langle\rho_a,a| X_0 E |\rho_b,b \rangle}{ \langle\rho_a,a| \rho_b,b \rangle} &= - b \frac{\langle\rho_a,a| X_0 |\rho_b,b \rangle}{ \langle\rho_a,a|\rho _b,b \rangle} \\ &\quad + (1- \rho_b) \frac{ d }{ d \rho} \biggl[ \biggl( \frac{\rho_a - \rho_b }{ \rho_a - \rho} \biggr)^{a+b} \frac{\langle\rho_a,a| X_0 |\rho,b \rangle}{ \langle\rho_a,a|\rho,b \rangle} \biggr] \bigg\vert _{\rho=\rho_b} . \end{aligned}$$

(66)

These last two formulae are exact and valid for all values of ρ _a ,ρ _b ,a,b. (They have been derived from (61) under the condition that ρ _a >ρ _b and 0<b<1, but as all expressions are rational functions of all their arguments, they remain valid everywhere.)

From (66) it is possible to show that Φ(μ,h) defined by

$$ \varPhi(\mu,h) = \frac{ \langle W | \exp[ (e^h D + E) \mu] | V \rangle }{ \langle W | V \rangle} $$

(67)

satisfies the following equation

$$\frac{d \varPhi}{ d \mu}= \frac{ b(1+ \rho_a (e^h-1)) + a (1+ \rho_b (e^h-1)) }{ \rho_a-\rho_b} \varPhi+ \bigl(1+\rho_b \bigl(e^h-1\bigr)\bigr) \frac{ d \varPhi}{ d\rho_b} . $$

This equation can be solved by the method of characteristics, which tells us that the solution is of the form

The fact that Φ(0,h)=1 determines the unknown function and one gets

$$\begin{aligned} \begin{aligned}[b] \varPhi(\mu,h) &= \biggl( \frac{(\rho_a- \rho_b) ( e^h -1) }{ 1+ \rho_a (e^h-1) - \exp[\mu(e^h-1)] ( 1 + \rho_b(e^h-1)) } \biggr)^{a+b}\\ &\quad \times \exp\bigl[b \mu\bigl(e^h-1\bigr) \bigr] \end{aligned} \end{aligned}$$

(68)

(see Eqs. (3.7)–(3.10) of [13]).

This expression becomes singular as μ→μ ₀ with

$$\mu_0=\frac{1 }{ e^h-1}\log \biggl(\frac{1+ \rho_a(e^h-1) }{ 1+ \rho _b(e^h-1)} \biggr) $$

and by analysing the power law singularity one can get the asymptotic expression valid for large L

$$ \dfrac{\langle W | (e^h D + E)^L | V \rangle }{ \langle W | V \rangle } \simeq \dfrac{\varGamma(a+b+ L) (\rho_a-\rho_b)^{a+b} \ \mu _0^{-L-a-b} }{ \varGamma(a+b) \ (1+\rho_a(e^h-1))^a \ (1+\rho_b(e^h-1))^b } . $$

(69)

Second consequence of (61)

Another important consequence which can be obtained by dividing (61) by (62) is the following additivity formula

$$\begin{aligned} & \frac{\langle\rho_a,a|X_1 X_2|\rho_b,b\rangle}{ \langle\rho_a,a| (D+E)^{L+L'} | \rho_b,b\rangle} \\ &\quad = \frac{\varGamma(L+a+b) \varGamma(L'+1) }{ \varGamma(L+L'+a+b)} \oint_{\rho_b<|\rho|<\rho_a} \frac{d\rho}{2i\pi} \\ &\qquad\times \frac{(\rho_a-\rho_b)^{a+b+L+L'}}{(\rho_a-\rho)^{a+b+L}(\rho-\rho_b)^{1+L'}} \frac{\langle\rho_a,a|X_1|\rho, b\rangle}{ \langle\rho_a,a| (D+E)^L | \rho, b\rangle} \frac{\langle\rho,1-b|X_2|\rho_b,b\rangle}{ \langle\rho,1-b| (D+E)^{L'} | \rho_b,b\rangle} . \end{aligned}$$

(70)

which is the same as Eq. (65) of [3] up to the prefactor which was wrong in [3] and which is corrected here. This formula allows one to compute the properties of a lattice of L+L′ sites if one knows those of two systems of size L and L′.

Third consequence of (61)

Using (66) and (62) one can write an exact recursion for Z _L defined in (2)

$$ Z_{L+1}= \biggl[1+ \rho_b \ e^{h_{L+1}} + b \frac{\rho_a-\rho_b }{ L+a+b} \ e^{h_{L+1}} \biggr] Z_L + \frac{ (\rho_a-\rho_b) (1+ \rho_b \ e^{h_{L+1}} ) }{ L+a+b} \frac{d Z_L }{ d \rho_b} . $$

(71)

We won’t use this recursion relation in this paper, but we believe that it could be an alternative starting point to recover the result (9) and possibly further corrections.

Appendix B: Derivation of (51), (52), (53) in the equilibrium case

Let us consider a site dependent field h _i with small variations

$$z_i= h_i - h $$

around a certain value h. One can then expand G _L defined in (2), (3) in powers of the z _i ’s

$$G_L(h_1, \ldots, h_L) = G_L(h , \ldots, h ) + \sum_i z_i \langle n_i\rangle + \frac{1 }{ 2} \sum_{i , j} z_i z_j \langle n_i n_j \rangle_c + O \bigl(z^3 \bigr) $$

where 〈.〉 denotes an average in the constant field h. Far from the boundaries i.e. when i≫1 and L−i≫1, the correlations become translational invariant (because the system is at equilibrium)

$$ \langle n_i \rangle= g'(h ) ;\qquad \langle n_i n_j \rangle_c = c_{j-i}(h ) $$

(72)

and

$$g''(h)= \sum_{k=-\infty}^\infty c_k(h) . $$

One can rewrite G _L (h ₁,…,h _L ) as

$$\begin{aligned} G_L(h_1, \ldots, h_L) =& G_L(h , \ldots, h ) + g'(h ) \sum _i z_i + \frac{c_0(h ) }{ 2 } \sum _i z_i^2 + \sum _{k \ge1} c_k(h ) \sum_{i=1 }^{L-k} z_i \, z_{i+k} \\ &{}+\sum_i z_i \bigl(\langle n_i\rangle - g'(h ) \bigr)+ \frac{1 }{ 2} \sum _{i , j} z_i z_j \bigl( \langle n_i n_j \rangle_c -c_{j-i}(h ) \bigr) \\ &{}+ O \bigl(z^3 \bigr) \end{aligned}$$

(73)

and using the fact (which follows from (73) by looking at the term proportional to L when all the h _i ’s are equal) that

$$g(h_i)= g(h ) + z_i\, g'(h ) + \frac{z_i^2 }{ 2} \sum_{k =-\infty} ^\infty c_k(h )+ O \bigl(z^3 \bigr) $$

one gets

$$\begin{aligned} & G_L(h_1, \ldots, h_L) - \sum _i g(h_i) \\ &{}\quad = G_L(h, \ldots, h) - L g(h) \\ &{}\qquad + \sum_{k \ge1} c_k(h ) \Biggl[ \sum_{i=1 }^{L-k} \biggl( z_i z_{i+k}-\frac{z_i^2 + z_{i+k}^2 }{ 2} \biggr) -\frac{1 }{ 2} \sum_{i= 1}^k z_i^2 - \frac{1 }{ 2} \sum_{i= L-k+1 }^L z_i^2 \Biggr] \\ &{}\qquad +\sum_i z_i \bigl( \langle n_i\rangle - g'(h ) \bigr)+ \frac{1 }{ 2} \sum_{i , j} z_i z_j \bigl( \langle n_i n_j \rangle_c -c_{j-i}(h ) \bigr) + O \bigl(z^3 \bigr) . \end{aligned}$$

(74)

In the large L limit, the correlation functions, near the boundaries, have a limit which is not translational invariant

$$\langle n_i \rangle- g'(h ) \to a_i^{\rm left }(h ) ;\qquad \langle n_{L-i} \rangle- g'(h ) \to a_i^{\rm right}(h ) $$

whereas

$$\langle n_i \, n_j \rangle_c- c_{j-i}(h ) \to b_{i,j}^{\rm left }(h ) ; \qquad \langle n_{L-i} \, n_{L-j} \rangle_c- c_{j-i}(h ) \to b_{i,j}^{\rm right }(h ) . $$

One then should have

$$ \frac{d A^{\rm left} (h) }{ dh} = \sum_{i=1}^{\infty} a_i^{\rm left }(h) ; \qquad \frac{d A^{\rm right} (h) }{ dh} = \sum _{i=0}^{\infty} a_i^{\rm right }(h) , $$

(75)

$$ \frac{d a_i^ {\rm left } (h) }{ dh} = \sum_{j} b_{i,j}^{\rm left }(h) - \sum_{k \ge i} c_{k}(h) ;\qquad \frac{d a_i^ {\rm right } (h) }{ dh} = \sum _{j} b_{i,j}^{\rm right }(h) - \sum _{k \ge{i+1}} c_{k}(h) $$

(76)

so that using (46) and the fact that c _k (h)=c _−k (h)

$$\frac{d^2 A^{\rm left} (h) }{ dh^2} = \sum_{i\ge1,j \ge1} b_{i,j}(h) - \sum_{k \ge1} k \, c_{k}(h) ; \qquad \frac{d^2 A^{\rm right} (h) }{ dh^2} = \sum _{i\ge0,j \ge0} b_{i,j}(h) - \sum _{k \ge1} k\, c_{k}(h) . $$

For large L this becomes

$$\begin{aligned} & G_L(h_1, \ldots, h_L) - \sum _i g(h_i) \\ &\quad = A^{\rm left}(h) + A^{\rm right}(h) \\ & \qquad + \sum_{k \ge1} c_k(h ) \Biggl[ \sum_{i=1 }^{L-k} \biggl( z_i z_{i+k}-\frac{z_i^2 + z_{i+k}^2 }{ 2} \biggr) -\frac{1 }{ 2} \sum _{i= 1}^k z_i^2 - \frac{1 }{ 2} \sum_{i= L-k+1 }^L z_i^2 \Biggr] \\ & \qquad +\sum_i z_i \bigl(\langle n_i\rangle - g'(h ) \bigr)+ \frac{1 }{ 2} \sum _{i , j} z_i z_j \bigl( \langle n_i n_j \rangle_c -c_{j-i}(h ) \bigr) + O \bigl(z^3 \bigr) \end{aligned}$$

which can be rewritten, up to terms of third order in the z _i ’s

$$ G_L(h_1, \ldots, h_L) - \sum _i g(h_i) = D^{\rm left} + D^{\rm right}- \sum_{k \ge1} c_k(h_i) \Biggl[ \sum_{i=1 }^{L-k} \frac{ ( h_i- h_{i+k})^2 }{ 2} \Biggr] $$

(77)

where

$$\begin{aligned} D^{\rm left} =& A^{\rm left} (h_1) - \frac{1 }{ 2} \sum_{k \ge1} c_k(h_1) \Biggl[ \sum_{i=1 }^k (h_i-h_1)^2 \Biggr] +\sum_i (h_i-h_1) \, a_i^{\rm left }(h_1) \\ &{} + \frac{1 }{ 2} \sum_{i , j} (h_i-h_1) (h_j-h_1)\, b_{i,j}^{\rm left }(h_1) + O \bigl(z^3 \bigr) \end{aligned}$$

and

$$\begin{aligned} D^{\rm right} =& A^{\rm right} (h_L) - \frac{1 }{ 2} \sum_{k \ge1} c_k(h_L) \Biggl[ \sum_{i=1 }^k (h_{L+1-i}-h_L)^2 \Biggr] +\sum_i (h_{L-i} -h_L)\, a_i^{\rm right}(h_L) \\ &{} + \frac{1 }{ 2} \sum_{i , j} (h_{L-i}-h_L) (h_{L-j}-h_L) \, b_{i,j}^{\rm right}(h_L) + O \bigl(z^3 \bigr) . \end{aligned}$$

All the differences h _i −h _j which appear in (77) are between nearby sites i,j. Under this form, the differences h _i −h _j between remote sites do not need to be small. In what follows we will assume that (77) remains true as long as these differences h _i −h _j remain small for nearby sites (i.e. for |i−j|≪λ) even if these differences could be large when |i−j|∼λ.

Now for a slowly varying field of the form (4), with L=yλ as in (5) one can evaluate the different terms using the Euler MacLaurin formula

$$\begin{aligned} &\sum_{i=1}^L g(h_i) \simeq \lambda \int_0^y g\bigl(H(x)\bigr) dx - \frac{ H'(y )g'(H(y )) - H'(0) g'(H(0)) }{ 24 \lambda} , \\ & D^{\rm left} \simeq A^{\rm left} \bigl(H(0)\bigr) + \frac{H'(0) }{ \lambda} \sum_{i \ge1} \biggl( i -\frac{1}{ 2} \biggr) a_i^{\rm left }\bigl(H(0)\bigr) , \\ & D^{\rm right} \simeq A^{\rm right} \bigl(H (y ) \bigr) - \frac{H' (y ) }{ \lambda} \sum_{i \ge0} \biggl( i + \frac {1}{ 2} \biggr) a_i^{\rm right } \bigl(H (y ) \bigr) , \\ & \sum_{k \ge1} c_k(h_i) \Biggl[ \sum_{i=1 }^{L-k} \frac{ ( h_i- h_{i+k})^2 }{ 2} \Biggr] \simeq\frac{1 }{ 2\lambda} \int_0^{y } \sum _{k \ge1} k^2\, c_k\bigl(H(x) \bigr) H'(x)^2 dx . \end{aligned}$$