Finite Size Scaling of the Dynamical Free-Energy in a Kinetically Constrained Model

Abstract

We determine the finite size corrections to the large deviation function of the activity in a kinetically constrained model (the Fredrickson-Andersen model in one dimension), in the regime of dynamical phase coexistence. Numerical results agree with an effective model where the boundary between active and inactive regions is described by a Brownian interface.

Appendices

Appendix A: Scaling of ψ _L (s) in the Interface Model

We determine in this appendix the finite size corrections to the function $\hat{\varphi}_{L}(\lambda)$, defined in (9), in the effective interface description discussed in part 2.2. Let us first focus on a system with one boundary x(t) between the active and inactive regions. It performs a random walk of jump rate p (resp. q) to the left (resp. right), starting from x=0 at time 0, and constrained to come back to 0 at final time t. The walk x(t) is reflected at 0 so that x(t)≥0. Appropriate values of p and q are discussed in Sect. 4.1. The computation is done for generic values of p and q.

In the original model, the cost of maintaining the interface (that is, for x(τ) to come back in 0 at time τ=t) is given by the surface tension Σ. In this effective description

$$Z_{\mathrm{eff}}(s,t)\equiv\frac{ \langle e^{-s{\mathbb{K}}\int_0^td\tau\,x(\tau)}\delta (x(t)=0) \rangle_{p,q}}{ \langle\delta(x(t)=0) \rangle_{p,q}}$$

(43)

where ${\mathbb{K}}= 4 c^{2}(1-c)$ is the mean density of activity and 〈⋅〉 _p,q denotes the average on trajectories x(τ)_0≤τ≤t without constraint at final time. The normalisation 〈δ(x(t)=0)〉 _p,q is e.g. fixed from Z _eff(0,t)=1.

Let us denote by P(x,X,t) the probability of being in x at time t, having observed a value X of the area $\int_{0}^{t} d\tau\,x(\tau)$. The initial condition is P(x,X,0)=δ _x,0 δ(X). Defining $\hat{P}(x,s,t)=\int dX\,e^{-s{\mathbb{K}}X} P(x,X,t)$, one has (using the constraint at final time x(t)=0):

$$Z_{\mathrm{eff}}(s,t) = \frac{\hat{P}(0,s,t)}{\hat{P}(0,0,t)}$$

(44)

Moreover, from Feynman-Kac formula, the time evolution of $\hat{P}(x,s,t)$ is given by

(45)

and a reflection at x=0. To symmetrise the walk, we now set $\hat{Q}(x,s,t)=(q/p)^{\frac{x}{2}}\hat{P}(x,s,t)$. One has again $Z_{\mathrm{eff}}(s,t) = \frac{\hat{Q}(0,s,t)}{\hat{Q}(0,0,t)}$ and the evolution of $\hat{Q}(x,s,t)$ writes

(46)

The normalisation has changed but we see that apart from the constant loss rate $(2\sqrt{pq}-p-q )$, $\hat{Q}$ describes a symmetric walk with the same term $s{\mathbb{K}}x\hat{Q}(x,s,t)$ corresponding to weighting trajectories by the area $\int_{0}^{t} d\tau\,x(\tau)$. In other words, defining at last $\tilde{Q}(x,s,t) = \hat{Q}(x,s,t) e^{-t(2\sqrt{pq}-p-q)}$, one has

$$Z_{\mathrm{eff}}(s,t) = \frac{\tilde{Q}(0,s,t)}{\tilde{Q}(0,0,t)}$$

(47)

and from the equation of evolution

(48)

we see that $\tilde{Q}(0,0,t)$ does not increase exponentially in time since for s=0 the equation describes a simple symmetric random walk. Moreover, in the large size limit (s→0), the evolution of $\tilde{Q}(x,s,t)$ is governed by the continuous in space operator

$$\sqrt{pq}\partial_x^2 - s{\mathbb{K}}x = 2 \sqrt{pq}\biggl[ \frac{1}{2} \partial_x^2 - \frac{s{\mathbb{K}}}{2\sqrt{pq}} x\biggr]$$

(49)

with reflecting boundary condition in 0. For bridges, the spectrum is known and its largest eigenvalue is given by [23]

$$\psi^{\mbox{\scriptsize{non-per}}}_{\mathrm{eff}}(s)=2\sqrt{pq} \biggl(\frac {s{\mathbb{K}}}{2\sqrt{pq}} \biggr)^{\frac{2}{3}} 2^{-\frac{1}{3}}\alpha_1$$

(50)

where α ₁≈2.3381… is the first zero of the Airy function on the negative real axis. In periodic boundary conditions, one has two interfaces and the equivalent jump rates are multiplied by 2. Finally, this gives in the s→0 limit:

$$\psi^{\mathrm{per}}_{\mathrm{eff}}(s)=4\sqrt{pq} \biggl(\frac{s{\mathbb{K}}}{4\sqrt{pq}}\biggr)^{\frac{2}{3}} 2^{-\frac{1}{3}}\alpha_1$$

(51)

Appendix B: A Generic Identity Between Large Deviation Functions

In this appendix we prove an identity used in part 3.2 between large deviation functions associated to the activity K _t and to the escape rate R _t defined below. We consider a Markov process on a finite number of configurations $\{\mathcal{C}\}$, with transition rates $W(\mathcal{C}\to\mathcal{C}')$ between configurations. The activity K _t is a history-dependent observable increasing by 1 upon jumping from $\mathcal{C}$ to $\mathcal{C}'$. The probability $P(\mathcal{C},K,t)$ of being in $\mathcal{C}$ at time t having observed a value K of the observable K _t thus evolves in time through

$$\partial_t P(\mathcal{C},K,t) = \sum_{\mathcal{C}'}W\bigl(\mathcal{C}'\to\mathcal{C}\bigr) P\bigl(\mathcal{C}',K-1,t\bigr) - r(\mathcal{C}) P(\mathcal{C},K,t)$$

(52)

with $r(\mathcal{C})= \sum_{\mathcal{C}'} W(\mathcal{C}\to\mathcal{C}')$ the escape rate from configuration $\mathcal{C}$. The Laplace transform $P(\mathcal{C},s,t) =\sum_{K} e^{-sK}P(\mathcal{C},K,t)$ verifies

$$\partial_t P(\mathcal{C},s,t) = \sum_{\mathcal{C}'}e^{-s}W\bigl(\mathcal{C}'\to\mathcal{C}\bigr) P\bigl(\mathcal{C}',s,t\bigr) - r(\mathcal{C}) P(\mathcal{C},s,t)$$

(53)

The cumulant generating function ψ(s) defined in the infinite time limit as $\langle e^{-sK_{t}}\rangle\sim e^{t\psi(s)}$ is the largest eigenvalue of the operator $\mathbb{W}_{s}$ of elements

$$(\mathbb{W}_s )_{\mathcal{C}\mathcal{C}'}= e^{-s} W\bigl(\mathcal{C}'\to\mathcal{C}\bigr) - r(\mathcal{C})\delta_{\mathcal{C}\mathcal{C}'} $$

(54)

since $\langle e^{-sK_{t}}\rangle=\sum_{\mathcal{C}}P(\mathcal{C},s,t)$. Both K _t and $R_{t}=\int_{0}^{t}d\tau\,r(\mathcal{C}(\tau))$ quantify the activity of the histories. Their large deviation functions (ldf) are closely related. Indeed, let’s consider the joint ldf

$$\varPsi(s,\sigma)= \lim_{t\to\infty}\frac{1}{t} \log\bigl\langle e^{-sK_t-\sigma R_t}\bigr\rangle $$

(55)

As previously, one checks that Ψ(s,σ) is given by the maximum eigenvalue of the operator $\mathbb{W}_{s,\sigma}$ of elements [7]

$$(\mathbb{W}_{s,\sigma} )_{\mathcal{C}\mathcal{C}'}= e^{-s} W\bigl (\mathcal{C}'\to\mathcal{C}\bigr) - (1+\sigma)r(\mathcal{C})\delta_{\mathcal{C}\mathcal{C}'}$$

(56)

It verifies the symmetry $\mathbb{W}_{s,\sigma} = (1+\sigma)\mathbb{W}_{s+\log(1+\sigma),0}$ and so does the ldf:

$$\varPsi(s,\sigma)= (1+\sigma)\varPsi\bigl(s+\log(1+\sigma),0\bigr) $$

(57)

Besides, the mean values of K _t and R _t in the s-state are given by

$$\frac{1}{t} \langle K_t\rangle_s = -\partial_s\varPsi(s,\sigma) |_{\sigma=0},\qquad \frac{1}{t} \langle R_t\rangle_s = -\partial_\sigma\varPsi(s,\sigma) |_{\sigma=0}$$

(58)

Differentiating the symmetry (57) with respect to σ and sending σ to 0, one gets

$$\partial_\sigma\varPsi(s,\sigma) |_{\sigma=0} = \varPsi(s,0)+\partial_s\varPsi(s,\sigma) |_{\sigma=0}$$

(59)

which implies

$$\psi(s) = \frac{1}{t} \langle K_t\rangle_s - \frac{1}{t}\langle R_t\rangle_s $$

(60)

This relation is generic. It leads to the relation (14) between scaling exponents for the FA model in the inactive regime s=λ/L.

Appendix C: Bernoulli Approximation to Determine φ _L (λ)

In this appendix, we obtain the expression of the Courant-Fischer optimisation principle of part 3.3 for Bernoulli states. Before this, one needs to symmetrise the evolution operator $\mathbb{W}_{s}$ introduced in (54). We take the notation of Appendix B. Assuming that the jump rates verify the detailed balance symmetry $W(\mathcal{C}\to\mathcal{C}')P_{\mathrm{eq}}(\mathcal{C})=W(\mathcal{C}'\to\mathcal{C})P_{\mathrm{eq}}(\mathcal{C}')$, it is generically possible to symmetrise the operator of evolution $\mathbb{W}_{s}$ through the similarity transformation $\mathbb{W}^{\mathrm{sym}}_{s}\equiv \hat{P}^{-\frac{1}{2}}_{\mathrm{eq}} \mathbb{W}_{s} \hat{P}^{\frac{1}{2}}_{\mathrm{eq}}$, where $\hat{P}_{\mathrm{eq}}$ is the diagonal operator of elements $P_{\mathrm{eq}}(\mathcal{C})$. Upon symmetrisation, we have that ψ(s) is also the largest eigenvalue of the symmetric operator $\mathbb{W}^{\mathrm{sym}}_{s}$ of elements

$$\bigl(\mathbb{W}^{\mathrm{sym}}_s \bigr)_{\mathcal{C}\mathcal{C}'}=e^{-s} \bigl[W\bigl(\mathcal{C}'\to\mathcal{C}\bigr) W\bigl(\mathcal{C}\to\mathcal{C}'\bigr) \bigr]^{\frac{1}{2}} - r(\mathcal{C})\delta_{\mathcal{C}\mathcal{C}'}$$

(61)

It is convenient to represent the operator of evolution in terms of spin $\frac{1}{2}$ operators σ ^± and $\hat{n}$. On each site i, σ ^± is the creation/annihilation operator and $\hat{n}$ is the counting operator. They are defined by

(62)

(63)

Where |0〉 and |1〉 are the vectors for empty and occupied states. One has

$$\mathbb{W}_s = \sum_{1\leq i\leq L}\bigl[e^{-s} \bigl(c\sigma^+_i+(1-c)\sigma_i^-\bigr)-c(1-\hat{n}_i)-(1-c)\hat{n}_i \bigr](\hat{n}_{i+1}+\hat{n}_{i-1})$$

(64)

Transition rates obey detailed balance with respect to the product Bernoulli measure of uniform density c (conditioned to exclude the fully inactive configuration). The symmetrised operator of evolution writes

$$\mathbb{W}^{\mathrm{sym}}_s = \sum_{1\leq i\leq L}\bigl[e^{-s}\sqrt{c(1-c)}\bigl(\sigma^+_i+\sigma_i^-\bigr)-c(1-\hat{n}_i)-(1-c)\hat{n}_i \bigr](\hat{n}_{i+1}+\hat{n}_{i-1})$$

(65)

Defining the vector |ρ〉 corresponding to the Bernoulli distribution of density $\sqrt{\rho}$:

$$|\rho\rangle= \sqrt{\rho} |1\rangle+ (1-\sqrt{\rho}\,) |1\rangle $$

(66)

and taking |X〉 in (17) to be the state |ρ ₁…ρ _L 〉^cond conditioned to exclude the fully empty configuration:

$$|\rho_1\ldots\rho_L\rangle^{\mathrm{cond}} = \sum _{n_i:\sum_in_i\neq0}\langle n_1\ldots n_L|\rho_1 \ldots\rho_L\rangle |n_1\ldots n_L\rangle $$

(67)

one obtains (18) by direct computation.