Finite-size and finite-time effects in large deviation functionsnear dynamical symmetry breaking transitions

# Finite-size and finite-time effects in large deviation functions near dynamical symmetry breaking transitions

Yongjoo Baek DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom    Yariv Kafri Department of Physics, Technion, Haifa 32000, Israel    Vivien Lecomte UniversitÃ© Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France
###### Abstract

We introduce and study a class of particle hopping models consisting of a single box coupled to a pair of reservoirs. Despite being zero-dimensional, in the limit of large particle number and long observation time, the current and activity large deviation functions of the models can exhibit symmetry-breaking dynamical phase transitions. We characterize exactly the critical properties of these transitions, showing them to be direct analogues of previously studied phase transitions in extended systems. The simplicity of the model allows us to study features of dynamical phase transitions which are not readily accessible for extended systems. In particular, we quantify finite-size and finite-time scaling exponents using both numerical and theoretical arguments. Importantly, we identify an analogue of critical slowing near symmetry breaking transitions and suggest how this can be used in the numerical studies of large deviations. All of our results are also expected to hold for extended systems.

## I Introduction

In recent years, there has been much interest in large deviation functions (LDFs, see Touchette (2009) for a review) encoding the probability of atypical fluctuations in time-averaged observables of many-body quantum Levitov and Lesovik (1993); Levitov et al. (1996); Pilgram et al. (2003); Jordan et al. (2004); Dereziński et al. (2008); Esposito et al. (2009); Z̆nidaric̆ (2014); Genway et al. (2014); Carollo et al. (2017) and classical stochastic systems Derrida (2007); Derrida and Appert (1999); Derrida et al. (2004); Bodineau and Derrida (2004); Maes and Netočný (2008); Prolhac and Mallick (2008); Bodineau et al. (2008); Imparato et al. (2009); Lecomte et al. (2010); Prados et al. (2011); de Gier and Essler (2011); Lazarescu and Mallick (2011); Derrida (2011); Gorissen et al. (2012); Gorissen and Vanderzande (2012); Krapivsky and Meerson (2012); Flindt and Garrahan (2013); Akkermans et al. (2013); Meerson and Sasorov (2013); *MeersonPRE2014; Hurtado et al. (2014); Lazarescu (2015); Zarfaty and Meerson (2016). Of special interest have been LDFs of the time-averaged current and activity, the latter quantifying the mean frequency of dynamical events during a given observation period. Since both quantities are determined by the full history rather than the instantaneous state, even in thermal equilibrium, their LDFs can exhibit unexpected behaviors. In particular, even if the steady-state probability distribution of instantaneous quantities, such as the density profile of particles in the system, contains no singularities, the LDF of time-averaged quantities can be singular, giving rise to a dynamical phase transition (DPT). This happens since the dominant history leading to a given atypical time-averaged quantity can change in an abrupt way as the value of the time-averaged quantity is varied. Like equilibrium phase transitions, DPTs can occur as first, second, or even higher-order singularities of LDFs. To date, DPTs have been found in a host of systems encompassing driven diffusive systems Harris et al. (2005); Bertini et al. (2005, 2006); Bodineau and Derrida (2005, 2007); Appert-Rolland et al. (2008); Prolhac and Mallick (2009); Hurtado and Garrido (2009); *HurtadoPRL2011; Lecomte et al. (2012); Hirschberg et al. (2015); Jack et al. (2015); Baek et al. (2017, 2018); Shpielberg et al. (2017); Shpielberg (2017); Shpielberg et al. (2018), kinetically constrained models Garrahan et al. (2007, 2009); Bodineau et al. (2012); Nemoto et al. (2014, 2017), interface growth Majumdar and Schehr (2014); Le Doussal et al. (2016); Janas et al. (2016); Smith et al. (2018), and active particles Cagnetta et al. (2017); Nemoto et al. (2019).

Most of the DPTs have been obtained in many-body extended systems111See Speck et al. (2007); Tsobgni Nyawo and Touchette (2016); *TsobgniNyawoPRE2018; Garrahan et al. (2009) for exceptions. whose sizes are taken to be infinite. It is natural to ask how much of the observed phenomenology is related to the fact that these systems are extended. In this paper, we address this question by introducing a class of models consisting of a one-site (or single-box) system connected to a pair of reservoirs and studying their current and activity large deviations. Instead of taking a limit where the system size goes to infinity, we utilize a recently introduced formalism Baek et al. (2016) where , the maximum number of particles in the box, is arbitrarily large. Applying the saddle-point method, it is shown that even such models can exhibit DPTs induced by the breaking of the particle-hole symmetry, which was theoretically predicted Baek et al. (2017, 2018) and numerically observed Pérez-Espigares et al. (2018) in extended systems, with exactly the same critical exponents.

Importantly, the reduced dimensionality of a single-box model allows us to easily predict and confirm the effects of finite time, , and finite size, , on the critical phenomena near a symmetry-breaking DPT for arbitrary hopping rates. In previous studies of extended systems, finite-size scaling theories have been proposed for second and first-order DPTs of an exclusion process  Appert-Rolland et al. (2008); Shpielberg et al. (2018); Gorissen and Vanderzande (2011) as well as for kinetically constrained models Bodineau and Toninelli (2012); Bodineau et al. (2012); Nemoto et al. (2014, 2017). Much less is known about finite-time effects222As we will see, the LDF in the infinite-time limit is given by the maximum eigenvalue of a well-defined operator, while the finite-time behavior of the LDF involves more eigenvalues., with only a few results concerning diffusive Krapivsky et al. (2014) and super-diffusive Prolhac (2016) relaxations of density fluctuations far away from any DPTs. For symmetry-breaking DPTs in extended systems with open boundaries, Ref. Baek et al. (2018) used heuristic arguments to predict finite-time and finite-size scaling exponents. These, however, have not been verified. In this paper, based on studies of finite- saddle-point trajectories and an exact diagonalization of the transition matrix at finite , we identify both the finite- and finite- scaling exponents and propose a scaling form encompassing both. In particular, we are able to characterize in detail the different finite- scaling regimes. We find a regime where the initial condition strongly influence the LDF and, as one might expect, a late regime where the initial conditions do not play any role. The results show that, near a symmetry-breaking DPT, a phenomenon analogous to critical slowing appears. Namely, the relaxation of the system from a given initial condition becomes anomalously slow as the DPT is approached. This might be used to locate such DPTs in numerics Giardinà et al. (2006); Lecomte and Tailleur (2007); Tailleur and Lecomte (2009); Giardinà et al. (2011); Nemoto and Sasa (2014); Nemoto et al. (2016); Ray et al. (2018); Brewer et al. (2018); Pérez-Espigares and Hurtado () and possibly experiments by data collapse.

The paper is organized as follows. In Sec. II, we introduce the single-box models and present a path-integral representation of their statistics. In Sec. III, we discuss how the theory of symmetry-breaking DPTs and the associated critical behaviors can be derived using a saddle-point method in the joint limit and . In Sec. IV, based on both numerical diagonalization and theoretical arguments, we study finite-size and finite-time effects, allowing us to characterize the critical features of the DPT. Finally, we conclude in Sec. V.

## Ii Single-box models with particle-hole symmetry

In this section, we describe the general setup considered in our study. First we introduce a general class of single-box models. We then focus on a subclass of systems which obey a particle-hole symmetry. Then we formulate their coarse-grained descriptions for large . This allows us to studying their DPTs using saddle-point asymptotics.

### ii.1 General single-box models

We consider a single box, whose state is characterized by the number of particles inside. The box can hold at most particles () and is coupled to a pair of particle reservoirs. The left (right) reservoir is described as a box with a fixed number of particles (). The particles are exchanged with the left reservoir according to

 (1)

where () denotes the rate of hopping from the left (right) box to the right (left), see Fig. 1. Similarly, the exchange with the right reservoir is described by

 (2)

We are interested in the statistics of current and activity during a time interval . Defining the number () of rightward (leftward) hops across any of the two bonds connecting the reservoirs to the system, we have the time-averaged current per bond

 JT≡12T[MR(T)−ML(T)] (3)

and the time-averaged activity per bond

 KT≡12T[MR(T)+ML(T)]. (4)

The joint scaled cumulant generating function (CGF) for and is defined as

 (5)

where denotes the average over histories. Using standard methods, described in Appendix A, one can show that

 eTΨ(λ,μ)=⟨eT(λJT+μKT)⟩=∫D[n,^n]e−∫T0dt[^n˙n−Hλ,μ(n,^n)], (6)

with an effective Hamiltonian

 Hλ,μ(n,^n) ≡WR(¯na,n)[e^n+(μ+λ)/2−1]+WL(¯na,n)[e−^n+(μ−λ)/2−1] +WL(n,¯nb)[e^n+(μ−λ)/2−1]+WR(n,¯nb)[e−^n+(μ+λ)/2−1]. (7)

Here is a momentum (integrated along the imaginary axis) conjugate to , and the Lagrange multiplier () is a counting variable conjugate to ().

We are mainly interested in models presenting second-order singularities in the scaled CGF. As we show below, these naturally occur for a class of models whose dynamics obey a particle-hole symmetry. For simplicity, we first consider the case where the two reservoirs have equal densities , which captures all the essential physics of the DPT. The generalization to the boundary-driven case is discussed in Appendix B.

### ii.2 Particle-hole symmetric models

The particle-hole symmetry is implemented by choosing a dynamics which is invariant under the combined operation of the particle-hole exchange and the exchange of the reservoir locations. This is achieved by imposing

 WR(n1,n2)=WR(N−n2,N−n1),WL(n1,n2)=WL(N−n2,N−n1). (8)

As stated above, we focus on the case where the reservoir densities are . We also assume that each hopping across a bond obeys local detailed balance, so that the rate of a rightward hop and that of a leftward one differ only due to a global field (bulk drive):

 WR(n1,n2)WL(n2,n1)=ν. (9)

Here controls the strength of the field. To simplify the notation, we write the rate of a rightward hop from the left reservoir into the box as

 WR(N2,n)=αV(n). (10)

Then, using Eqs. (8), (9), and (10), the four hopping rates in Eqs. (1) and (2) can be written as

 WR(N2,n) =αV(n), WL(N2,n) =ανV(N−n), WL(n,N2) =ανV(n), WR(n,N2) =αV(N−n). (11)

We note that, to impose the bound , the hopping rates are further constrained by

 V(N)=0. (12)

With these choices, the Hamiltonian in Eq. (II.1) takes the form

 Hλ,μ(n,^n) =α[ν+1ν(e^n+μ/2coshλ2−1)+ν−1νe^n+μ/2sinhλ2]V(n) +α[ν+1ν(e−^n+μ/2coshλ2−1)+ν−1νe−^n+μ/2sinhλ2]V(N−n), (13)

which can be rewritten as

 Hλ,μ(n,^n) =γ[(ze^n−1)V(n)+(ze−^n−1)V(N−n)]. (14)

Here we used definitions and

 z(λ,μ)≡eμ/2cosh(λ2+tanh−1ν−1ν+1)cosh(tanh−1ν−1ν+1). (15)

We note that the unbiased state corresponds to . From Eqs. (6), (14), and (15), one observes that the scaled CGF depends on and only through . We also note that satisfies

 z(λ,μ)=z(−λ−4tanh−1ν−1ν+1,μ), (16)

which reflects the Gallavotti–Cohen symmetry Gallavotti and Cohen (1995a); *GallavottiJSP1995.

So far we have described the microscopic dynamics in the sense that the discrete nature of the particles is maintained. We next formulate a coarse-grained description of the dynamics for large , which makes the models easier to study by changing to continuous state variables and facilitating saddle-point techniques.

### ii.3 Coarse-grained description for large N

To take the large- limit, it is useful to define the rescaled fields and introduce the rescaled time and observables

 n →Nρ, ^n →^ρ, V(n) →Nkv(ρ), t →N1−kt, JT →NkJT, KT →NkKT, (17)

where is a positive number determined by the structure of the hopping rates (see below for examples). We note that the constraint (12) can now be written as

 v(1)=0, (18)

which ensures . Using these in Eqs. (6) and (14), we obtain a rescaled path-integral representation for the scaled CGF , namely

 eNTψ(z)=⟨eNT(λJT+μKT)⟩=∫D[ρ,^ρ]e−NSz[ρ,^ρ] (19)

with the action

 Sz[ρ,^ρ]≡∫T0dt[^ρ˙ρ−Hz(ρ,^ρ)], (20)

where the Hamiltonian is given by

 Hz(ρ,^ρ) ≡γ[(ze^ρ−1)v(ρ)+(ze−^ρ−1)v(1−ρ)]. (21)

The particle-hole symmetry of the system is reflected in the symmetry of the action

 Sz[ρ,^ρ]=Sz[1−ρ,−^ρ]. (22)

For , from Eqs. (19), (20), and (21), we find that can be obtained by a saddle-point asymptotics

 ψ(z)=−infρ,^ρlimT→∞1TSz[ρ,^ρ], (23)

where the minimum action is achieved by real-valued and obeying the Hamiltonian dynamics

 ˙ρ=∂Hz∂^ρ =γz[v(ρ)e^ρ−v(1−ρ)e−^ρ], (24) ˙^ρ=−∂Hz∂ρ =−γ[v′(ρ)(ze^ρ−1)−v′(1−ρ)(ze−^ρ−1)]. (25)

Although is defined only in the limit, the above saddle-point trajectories still describe the histories dominantly contributing to the finite-time scaled CGF

 ψT(z)=−infρ,^ρ1TSz[ρ,^ρ] (26)

whenever is large.

## Iii Symmetry-breaking dynamical phase transitions

We now calculate the scaled CGF of the single-box model and show that, with a proper choice of rates, the model displays the same DPTs exhibited by extended systems. In particular, we are interested in the DPTs between a particle-hole symmetric phase and one where the symmetry is broken.

### iii.1 Particle-hole symmetric phase

It is easy to see that, for any and ,

 ρ=1/2,^ρ=0 (27)

yields a time-independent, particle-hole symmetric solution for Eqs. (24) and (25). If this symmetric saddle-point profile truly minimizes the action, Eq. (23) implies

 ψ(z)=ψsym(z) ≡−limT→∞1TSz[ρ(t)=1/2,^ρ(t)=0]=2γ¯v(z−1). (28)

Note that from here on we use the shorthand notations

 ¯v=v(1/2),¯v′=v′(1/2),¯v′′=v′′(1/2),¯v(n)=v(n)(1/2) for n≥3. (29)

In Appendix C, we discuss the condition for the symmetric solution in Eq. (27) to be the dominant profile in the unbiased state . We find that being a monotonically decreasing function of is a sufficient condition. We also note that the mean current and activity are obtained from the above relations as

 ⟨J⟩=∂λψ(λ,μ)|λ=μ=0=γ¯vν−1ν+1,⟨K⟩=∂μψ(λ,μ)∣∣λ=μ=0=γ¯v. (30)

A second-order DPT occurs when this symmetric solution becomes unstable with respect to small fluctuations as the value of is changed. To this end, in the next section we study the Gaussian fluctuations of the action.

### iii.2 Stability analysis

The fluctuations of the action around the symmetric saddle-point solution (27),

 Sz[1/2+φ(t),i^φ(t)]=Sz[1/2,0]+δ2Sz[φ,^φ]+O(φ3,^φφ2,^φ2φ,^φ3), (31)

are described by the Gaussian action

 δ2Sz[φ,^φ] =∫dt[i^φ∂tφ+γz¯v^φ2−2iγz¯v′^φφ+γ(1−z)¯v′′φ2] =γ∫dω2π[φω^φω]⎡⎢⎣2(1−z)¯v′′−ω+4iz¯v′2ω−4iz¯v′22z¯v⎤⎥⎦M[φ−ω^φ−ω], (32)

where and are Fourier transforms of and defined as

 φω≡∫dtφ(t)e−iωt,^φω≡∫dt^φ(t)e−iωt. (33)

The eigenvalues of for the typical state are given by , so that the symmetric solution is always stable in this case. As moves away from , the profile becomes unstable if

 detM=4[(¯v′)2−¯v¯v′′]z2+4¯v¯v′′z+ω24=0, (34)

whose roots are given by

 z=z∗±=2¯v¯v′′±√4¯v2(¯v′′)2+ω2[¯v¯v′′−(¯v′)2]4[¯v¯v′′−(¯v′)2]. (35)

For a DPT to occur, at least one of the roots should be real and positive. If this is the case, there are two possible scenarios:

1. Case of . This case requires , and the only positive root is

 z∗+=2¯v¯v′′+√4¯v2(¯v′′)2+ω2[¯v¯v′′−(¯v′)2]4[¯v¯v′′−(¯v′)2], (36)

which is always greater than and reaches the minimum at . Thus a DPT occurs due to a time-independent mode at

 z=zc=¯v¯v′′¯v¯v′′−¯v′2, (37)

which is always greater than . Revisiting Eq. (15), this implies that the symmetric (symmetry-broken) phase occupies the low-activity, low-current (high-activity, high-current) regime. A phase diagram in the -plane corresponding to this scenario is shown in Fig. 2(a). As will be shown later, a DPT between these two phases occurs as a second-order singularity of shown in Fig. 2(b), with the optimal density minimizing the action exhibiting clear bifurcations shown in Fig. 2(c) and corresponding to the symmetry breaking.

2. Case of . Here a positive root exists if and only if . It is then given by

 z∗−=−2¯v¯v′′+√4¯v2(¯v′′)2−ω2[(¯v′)2−¯v¯v′′]4[(¯v′)2−¯v¯v′′], (38)

which is always less than and reaches its maximal value at . Again, a DPT occurs due to a time-independent mode at given by Eq. (37), which satisfies . Combining this with Eq. (15), we find that the symmetric (symmetry-broken) phase occupies the high-activity, high-current (low-activity, low-current) regime. A phase diagram in the -plane for this scenario is illustrated in Fig. 2(d), with second-order singularities of and the optimal density shown in Fig. 2(e,f).

We note that while scenario has been observed before in extended systems Hurtado and Garrido (2011); Baek et al. (2017, 2018), we are not aware of any example of scenario , although it bears some similarities to the DPTs of the WASEP with open boundaries Baek et al. (2017, 2018); Pérez-Espigares et al. (2018) if one shifts and appropriately. In all scenarios, a symmetry-breaking DPT occurs due to a time-independent mode.

We next derive a Landau theory from first principles to describe the nature of the DPT in detail.

### iii.3 Exact Landau theory for dynamical phase transitions

Having shown that the DPTs are induced by time-independent modes, Eqs. (20) and (23) imply that the scaled CGF takes the form

 ψ(z)=supρ,^ρHz(ρ,^ρ)=ψ% sym(z)−infmLz(m), (39)

where and are time-independent solutions of Hamilton’s equations (24) and (25), and is an order parameter quantifying the broken particle-hole symmetry. In the vicinity of a DPT, where is of order , one can straightforwardly check that

 ρ=12+m,^ρ=−¯v′¯vm (40)

yields a time-independent solution of Eqs. (24) and (25) up to order . Using this solution in Eq. (39) and expanding in , we obtain

 Lz(m)=−aϵzm2+bm4 (41)

with the coefficients

 a≡γ¯v′′,b≡γ¯v′zc(14¯v′3¯v3+13¯v(3)¯v−12¯v′¯v′′¯v2−¯v′¯v(4)¯v¯v′′). (42)

The solution satisfies Eq. (39) up to order . This expression provides an exact Landau theory for the symmetry-breaking DPT near under the condition that — by tracking the optimal value of the order parameter minimizing , one observes a bifurcation of and an associated jump discontinuity of at (with the locations of symmetric and symmetry-broken phases determined by the sign of , as discussed above), see Fig. 2. If , one needs to expand Eq. (39) to higher order in . Note that, depending on the sign of , both scenario  and scenario  described in Sec. III.2 are captured by the Landau theory.

The Landau theory obtained above has the same form as the one describing symmetry-breaking DPTs in extended systems Baek et al. (2017, 2018). Thus the universal features of such DPTs are captured by our large- single-box models, whose only degree of freedom plays the role of the largest-wavelength mode in extended systems. Below we explicitly construct a single-box model motivated by the Katz–Lebowitz–Spohn (KLS) model Katz et al. (1984) which illustrates the phenomenology described so far.

Next, we examine the statistics of finite-frequency modes, which contains crucial information about the relaxation of the system near the transition. In particular, we find a behavior analogous to critical slowing down.

### iii.4 Critical slowing down

Let us define . In the symmetric phase (for ), from Eqs. (19), (31), and (III.2), we find that the Gaussian fluctuations around are characterized by the probability distribution

 Pz[φ]=∫D^φe−Nδ2Sz[φ,^φ]∼exp[−Nγ8¯vz∫dω2π(ω2+τ−2z)φωφ−ω], (43)

where

 τz≡√116¯vz|¯v′′ϵz|∼|ϵz|−1/2 (44)

has dimension of time. In the frequency space, the variance of the above distribution is given by

 ⟨φωφω′⟩z=8π¯vzNγ1ω2+τ−2zδ(ω+ω′), (45)

where denotes an average over the ensemble biased by . After applying the Fourier transform, the temporal correlations are obtained as

 ⟨φ(t)φ(t′)⟩z=2¯vzτzNγe−|t−t′|/τz. (46)

Thus is clearly interpreted as a correlation time, and its divergent behavior near a DPT implies critical slowing down. While this derivation is valid only in the symmetric phase, it is natural to expect that the same scaling behavior will still hold in the symmetry-breaking phase.

### iii.5 Example of symmetry breaking: Symmetric Antiferromagnetic Process

The KLS model is defined on a lattice where each site is occupied by at most one particle. The dynamics of the particles depend on nearest-neighbor interactions. Recently, it was shown that the KLS model, when connected to two reservoirs, exhibits a DPT when the interactions are sufficiently strongly antiferromagnetic Baek et al. (2017). In this case, the particles prefer a profile with only every second site occupied, which amounts to having a density . Then the noise strength in the dynamics is found to have a local minimum at . To mimic this behavior, we study a single-box model with the hopping rates

 WR(n1,n2) =n1(N−n2)[N2+ε4(n1+n2−N)2], WL(n1,n2) =n2(N−n1)[N2+ε4(n1+n2−N)2], (47)

with . These rates fulfill the conditions for the particle-hole symmetry and the bounded range of occupancy given in Eqs. (8) and (12). They also ensure that the hopping rate attains a local minimum when the two sites involved have an average occupancy . For this reason, we refer to this model as the Symmetric Antiferromagnetic Process (SAP).

For large , we can use Eqs. (9), (10), and (II.3) with to describe the model in terms of the rescaled parameters

 v(ρ)≡(1−ρ)[1+ε(ρ−12)2],α=12,ν=1. (48)

By Eqs. (15) and (37), we obtain

 z=eμ/2coshλ2,zc=εε−2. (49)

The corresponding Landau theory is derived from Eq. (41) as

 L(m)=−εϵzm2+2ε(1+ε)ε−2m4. (50)

Thus, if so that the coefficient of is positive, the model exhibits symmetry-breaking DPTs with the symmetry-broken phase occupying the high-current, high-activity regime. An example was already shown for in Fig. 2(a–c). We again stress that this Landau theory is a direct analogue of the one describing the symmetry-breaking DPT of the KLS model in extended systems.

Interestingly, if we generalize the model to negative values of (allowing the interactions to be ferromagnetic), the Landau theory predicts symmetry-breaking DPTs for as well. In this case, as illustrated for in Fig. 2(d–f), the symmetry-broken phase corresponds to the low-current, low-activity regime. For the sake of brevity, through the rest of this paper, we shall focus on the proper SAP with ; however, all the results we discuss below are also easily applicable to the DPTs for .

## Iv Effects of finite T or N

The simplicity of the single-box model provides a convenient avenue for addressing the effects of finite or on the symmetry-breaking DPTs, which are the main subject of this section. First, taking but leaving finite, we calculate analytically the optimal trajectory from a given initial state and show how its final point scales with as the system approaches a symmetry-breaking DPT. Second, we consider the case with finite and identify the exponents governing the finite- critical scalings near the DPT. These results allow us to build a comprehensive scaling theory near a symmetry-breaking DPT for finite and .

### iv.1 N→∞, finite T

#### iv.1.1 Formulation of the problem

Near a DPT we only need to consider trajectories which are close to the symmetric solution (27). With these considerations in mind, it is convenient to perform a canonical change of variables

 ρ=φ+12,^ρ=^φ−¯v′¯vφ. (51)

Since the transformation has a unit Jacobian, it does not introduce any additional term in the action. Thus, using Eqs. (20) and (21), the leading-order correction to the action arising from nonzero and is obtained as

 ΔSz[φ,^φ] ≡Sz[12+φ,−¯v′¯vφ+^φ]−Sz[12,0] =−¯v′¯vφ(T)2−φ(0)22+~Sz[φ,^φ], (52)

where

 ~Sz[φ,^φ]≡∫T0dt[^φ˙φ−h(φ,^φ)] (53)

with the effective Hamiltonian

 h(φ,^φ)≡Hz(12+φ,−¯v′¯vφ+^φ)−Hz(12,0). (54)

Our goal is to minimize for given values of and , the value of at time . In other words, we first find the action of the optimal Hamiltonian trajectory from to with the latter allowed to take any value; then, among all such trajectories, we choose the value of which gives the minimal action.

#### iv.1.2 Exact calculation of the optimal final point

To carry out the calculation of , we write the variations of for fixed and :

 δΔSz[φ,^φ]|φ(T)=∫T0dt{(˙φ−∂h∂^φ)δ^φ−(˙^φ+∂h∂φ)δφ}. (55)

This gives us as expected Hamilton’s equations

 ˙φ=∂h∂^φ,˙^φ=−∂h∂φ. (56)

Then, using Eq. (IV.1.1) and allowing variations of , we obtain

 δΔSz[φ,^φ]=[^φ(T)−¯v′¯vφ(T)]δφ(T)+δΔSz[φ,^φ]|φ(T). (57)

This implies that, among all the solutions of Eq. (56), the one with the minimal action satisfies

 ^φ(T)=¯v′¯vφ(T). (58)

To proceed, we note that the above relation gives a conserved “mechanical energy” of the Hamiltonian dynamics as a function of :

 E(φ(T))≡h(φ(T),¯v′¯vφ(T)). (59)

With this the minimum of can be written as

 infφ,^φΔSz[φ,^φ]=infφ(T)[−¯v′¯vφ(T)2−φ(0)22+∫φ(T)φ(0)dφ^φ − E(φ(T))T]. (60)

Differentiating the rhs with respect to and using Eq. (58), we find that the minimal requires

 ∫φ(T)φ(0)dφ∂^φ∂φ(T)−E′(φ(T))T=0. (61)

In the following discussions, the optimal is obtained by solving this equation.

#### iv.1.3 Numerical results for the SAP

With Eqs. (24), (25), and (61), we are ready to calculate the optimal finite- trajectories for given and . We first consider numerical solutions and identify different scaling regimes, each of which will be described by analytical arguments later. In Fig. 3, we illustrate such trajectories for the SAP with in the symmetry-broken phase, all of them starting from the initial state while the values of and are varied. The optimal trajectories themselves are marked by solid curves, whereas their final-time value  is shown as a dashed curves as changes continuously. Notably, if is sufficiently large, the trajectories initially appear to saturate at the value of the order parameter ; however, they eventually move past the plateau (with a characteristic time scale which, as shown below, reflects the critical slowing down ) and end up much closer to the symmetric state . As is evident from the data collapse, and exhibit different scaling behaviors near a DPT.

In Fig. 4, using the SAP with , we show that exhibits three different scaling regimes depending on the duration of the observation period :

• Regime I. If the observation period is not long enough, the initial state heavily influences the entire trajectory, including the final state obeying

 φ(T) ∼φ(0)/Tfor T≪φ(0)−1. (62)

The above scaling behavior is shown in Fig. 4(a).

• Regime II. As the observation period becomes longer, the initial-state dependence starts to disappear after a time scale , beyond which proximity to the critical point becomes manifest in the power-law decay

 φ(T) ∼T−2for φ(0)−1≪T≪|ϵz|−1/2, (63)

as also shown in the middle section of Fig. 4(b). At this stage, there is no distinction between the symmetric () and symmetry-broken () phases.

• Regime III. When is sufficiently larger than the correlation time scale , converges exponentially to zero in the symmetric phase (see Fig. 4(c)) and to nonzero values in the symmetry-broken phase (see Fig. 4(b)), as we show below:

 φ(T) ∼|ϵz|e−2√γ¯vzc|aϵz|T for T≫|ϵz|−1/2 and aϵz<0, limT→∞φ(T) ≃aϵz2√bc for T≫|ϵz|−1/2 and aϵz>0. (64)

Based on these scaling behaviors, one can infer the following scaling forms describing the crossovers between adjacent scaling regimes:

 φ(T) ={φ(0)2F1(Tφ(0))between regimes I and II,|ϵz|F2(T2aϵz)between % regimes II and III. (65)

To be consistent with the scaling behaviors in each regime, the functions and should satisfy

 F1(x) ∼{1/x for |x|≪1,1/x2 for |x|≫1, (66) F2(x) ∼⎧⎪⎨⎪⎩1/x for |z|≪1,e−const.×√|x| for |x|≫1 and x<0,const. for x≫1 and x>0. (67)

The existence of such () is manifest in the data collapse(s) shown in Fig. 4(a) (Fig. 4(b, c)).

Due to the simplicity of the single-box models, all the numerical results discussed above can be theoretically derived from first principles, as we now show.

#### iv.1.4 Derivation of the scaling theory

To analytically calculate satisfying Eq. (61), one needs to examine the form of the Hamiltonian . In what follows, we approximate by using Eq. (21) in Eq. (54) and expanding the latter for small and to obtain

 h(φ,^φ)=c^φ2−Lz(φ)+o(^φ2,ϵzφ2,φ4), (68)

where and are as defined in Eqs. (41) and (42), respectively. As we show, the results below are unaffected by the neglected higher-order terms. This approximate formula has a convenient interpretation as the Hamiltonian of a Newtonian particle of mass , velocity and position in an unstable quartic potential , represented schematically in Fig. 5.

Using Eqs. (54) and (59), the energy conservation implies

 ^φ=^φ±≡±√E(φ(T))+Lz(φ)γ¯vzc. (69)

Near a symmetry-breaking DPT, it is natural to expect that the optimal trajectory stays close to the symmetric solution (27). Thus the initial velocity should be in the uphill direction. For generic situations near the DPT, we expect to be well within the unstable branches of the potential (i.e., ), see Fig. 5. In this case, the sign of should be opposite to that of . Since the system satisfies a particle-hole symmetry, without loss of generality, we can focus on the case where , so that and .

Using the above relation and Eq. (61), we obtain

 2√γ¯vzcT≃∫φ(0)φ(T)dφ1√E(φ(T))+Lz(φ). (70)

This can be further simplified to

 2√γ¯vzcT≃∫φ(0)φ(T)dφ1√cφ(T)2−aϵzφ2+bφ4 (71)

by using Eq. (41) and noting that Eqs. (59) and (68) give

 E(φ(T))≃γ¯v′2zc¯vφ(T)2−Lz(φ(T))≃γ¯v′2zc¯vφ(T)2, (72)

where the second approximation is due to the quartic potential being negligible compared to the “kinetic” component near the DPT where . Depending on which term in the denominator dominates the integral in Eq. (71), we identify the following three scaling regimes in order of increasing :

Regime I. — Suppose that the integral in Eq. (71) is dominated by contributions from . Then, using a Taylor expansion, Eq. (71) can be approximated as

 T=φ(0)2γ¯v′zcφ(T)+O(ϵzφ(0)