# The Approach Towards Equilibrium in a Reversible Ising Dynamics Model: An Information-Theoretic Analysis Based on an Exact Solution

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## Abstract

We study the approach towards equilibrium in a dynamic Ising model, the Q2R cellular automaton, with microscopic reversibility and conserved energy for an infinite one-dimensional system. Starting from a low-entropy state with positive magnetisation, we investigate how the system approaches equilibrium characteristics given by statistical mechanics. We show that the magnetisation converges to zero exponentially. The reversibility of the dynamics implies that the entropy density of the microstates is conserved in the time evolution. Still, it appears as if equilibrium, with a higher entropy density is approached. In order to understand this process, we solve the dynamics by formally proving how the information-theoretic characteristics of the microstates develop over time. With this approach we can show that an estimate of the entropy density based on finite length statistics within microstates converges to the equilibrium entropy density. The process behind this apparent entropy increase is a dissipation of correlation information over increasing distances. It is shown that the average information-theoretic correlation length increases linearly in time, being equivalent to a corresponding increase in excess entropy.

## Keywords

Non-equilibrium Microscopic reversibility Entropy Ising model Information theory Excess entropy Q2R## 1 Introduction

The apparent contradiction between the reversibility of the microscopic equations of motions and the irreversibility of macroscopic processes has been a problem since the development of statistical mechanics by Maxwell, Boltzmann and Gibbs, see, e.g., refs. [6, 11]. How can microscopic reversibility be consistent with macroscopically irreversible phenomena like the second law of thermodynamics? This contradiction is often resolved by describing the approach to equilibrium in terms of coarse-graining of phase space or related approaches.

In this paper we take a microscopic perspective on the development of statistical properties of a system that follows a time evolution that is microscopically reversible. In what way can one understand how such a system “approaches” equilibrium? What is the role of internal correlations of the microstate and how do these change in the time evolution?

As an illustrative model we have chosen the energy conserving Ising dynamics model Q2R [14] in one dimension. We consider the system in the thermodynamic limit, i.e., an infinite sequence of spins, and it is assumed that the initial microstate is generated by a Bernoulli process with a dominating spin direction so that a magnetised and ordered (low entropy) configuration serves as the starting point for the dynamics.

The Q2R rule employs a parallel update according to a checkerboard pattern alternating between the white and black sites. This leads to a dynamics over a sequence of microstates, with (in general) changing internal statistical properties. Formally, we study how the dynamics changes the stochastic process that characterises the ensemble of microstates at the given time. The initial microstate is spatially ergodic, since it is a Bernoulli process. The same holds for any finite time step, even though a cellular automaton rule in general changes the process so that it becomes a hidden Markov model already after the first iteration.

We characterise the internal disorder (entropy) of a microstate at time *t* by the entropy density of the corresponding generating process. This entropy is also directly derived from the internal statistics of the microstate by taking into account all possible internal correlations. This can then be viewed as an internal measure of disorder of the microstate—a microscopic entropy [9].

Since the dynamics is microscopically reversible, the entropy density is conserved even if the stochastic process that generates the microstates changes [8]. The aim with this paper is to gain a full understanding on how this can be consistent with the apparent picture of a dynamics that brings the magnetised initial state of low entropy density into a state with zero magnetisation and a seemingly higher entropy density.

We solve exactly the dynamics of Q2R in one dimension, starting with a Bernoulli generated microstate, by deriving the statistical properties of the hidden Markov models that generate the microstates at any time *t*.

The picture that emerges is one where some correlations remain at short distance – in fact, exactly those that make sure that the energy is conserved. It is useful to discuss this in terms of ordered information, or negentropy (as the difference between full disorder and actual entropy density). This ordered information contains information in all correlations in the system, as well as density information, i.e., spin frequencies deviating from \(\{1/2,1/2\}\). Except for the nearest neighbour correlations, all other information is transferred to ever increasing distances. This leads to three observations: (i) the magnetisation quickly approaches zero, (ii) the local correlations approach those that characterise an equilibrium microstate at the given energy, (iii) the rest of the correlations (the negentropy) becomes more and more difficult to detect as they require larger and larger blocks of spins and their exact characteristics for their detection.

The focus of the present paper is to examine to what extent this process can be quantified, and whether we can make a more precise statement on how equilibrium is approached on the microscopic level.

In [13] microscopic reversibility and macroscopic irreversibility for the Q2R automaton was discussed looking at how the period length growth with the system size and thus showing that the recurrence time goes to infinity in the thermodynamic limit. In the present study we want to understand the approach to equilibrium from an information-theoretic point of view. The aim is to show and quantify how information in correlations are spread out over increasing distances so that, when observing configurations over shorter length scales, it appears as if the system is approaching equilibrium.

In [5] it is discussed in what way reversible and, more generally, surjective cellular automata exhibit mixing behaviour in the time evolution, i.e., whether there are cellular automata that in some way can be said to approach a random distribution (Bernoulli distribution with equal probabilities). The most well studied example is the XOR rule, see, e.g., [7, 8, 10], in which there is a randomization even though there are also recurrent, locally detectable, low entropy states, even for an infinite system. It is stated as an open question whether there are more physically relevant models that allow for a mathematical treatment of how such a mixing may occur, which then would imply a mixing modulo the energy constraint of the system, i.e., a maximization of the entropy density given the energy density [5]. We contribute to that question by providing the exact solution of the one-dimensional Q2R model as an example of a physically relevant model showing relaxation towards equilibrium.

The plan of the paper is the following: In Sect. 2 we introduce the model system—the Q2R cellular automaton—and discuss some of its known properties. In Sect. 3 we provide the analytical solution for the time evolution of a specific non-equilibrium probability distribution starting from independent spins in the one-dimensional Q2R cellular automaton. We use this solution to investigate the time evolution of the information-theoretic quantities and how they are consistent with the system achieving thermodynamic equilibrium. In particular we show that the correlation information can be divided into two different contributions—one part that reflects the equilibrium properties of the system (within interaction distance), and one part with an average correlation length that increases linearly in time. In Sect. 4, we discuss how the information-theoretic analysis explains how an equilibrium distribution is approached, even though the micro dynamics is reversible. The paper is then concluded by a discussion in Sect. 5.

## 2 Q2R: A Microscopically Reversible Ising Dynamics

We consider the Q2R model [14] in one dimension and in the limit of an infinite system. This means that we describe the spatial state (infinite sequence of spins) at a certain time as the outcome of a stationary stochastic process. The system is described as an infinite sequence of states, spin up or spin down, \(\uparrow \) and \(\downarrow \), respectively. In addition to this a state also holds the information whether to be updated or not in the current time step. The updating structure is such that every second spin is updated at *t*, and then at the next time step the other half of the lattice is updated, and so forth. The updating rule flips a spin when the spin flip does not change the energy, and it changes the state from updating to quiescent and vice versa. Normal nearest neighbour Ising interaction is assumed with an energy \(-1\) for parallel spins (\(\uparrow \uparrow \) or \(\downarrow \downarrow \)) and \(+1\) for anti-parallel spins (\(\uparrow \downarrow \) or \(\downarrow \uparrow \)). This means that the Q2R model is a micro canonical simulation of the Ising model, with conserved energy. It is also clear that the rule is reversible.

We assume that the initial state is generated by a Bernoulli process, and the aim is to give a statistical analysis of how spatial configurations change over time. Each time step is thus characterised by a certain stochastic process, and the Q2R rule transforms this process from one time step to the next.

Since the Q2R rule is reversible this implies that the entropy density *h*(*t*) of the ensemble at a given time step *t*, or, equivalently, the entropy rate of the stochastic process that generates the ensemble at time step *t*, is a conserved quantity under the Q2R dynamics. This follows, for example, from the observation that there is a local rule (also Q2R, but with a state shift) that runs backwards in time. Since both of these rules are deterministic and thus imply a non-increasing entropy density, the entropy density for an infinite system is conserved under Q2R.

Furthermore, we assume that the stochastic process is ergodic. Note that this is a spatial ergodicity, which implies that for almost all microstates in the ensemble (at any point in time), we have the sufficient statistics to calculate any information-theoretic properties depending on finite length subsystems of the microstate. This means that we can characterise a single microstate, at any time *t*, and identify its internal entropy density and correlation characteristics, which is identical to the same characteristics for the whole ensemble. This is conceptually appealing, since we can then identify an entropy density quantity as a property of a single microstate.

## 3 Analysis of the Time Evolution Starting from a Bernoulli Distribution

We assume that the initial spatial state is described by, or generated by, a Bernoulli process with probability \(0<p<1/2\) for spin up. In addition to this we augment our state variable with a second binary variable which marks every second lattice site being in an updating state (\(\underline{0}\) or \(\underline{1}\)), and the others in quiescent states (0 or 1), where the spin direction is denoted by 0 and 1 (with or without the underline mark) for spin up and down, respectively. Thus the spin up probability is \(p(\uparrow )=p(1)+p(\underline{1})\), and similarly for spin down.

*S*is the entropy of the probability distribution, as indicated by the equation.)

With an energy contribution from parallel and anti-parallel spins of \(-1\) and 1, respectively, we get the energy density \(u = -(1-2p)^2\) of the initial state. The system is not in equilibrium since the entropy density *h* is not in a maximum given the energy density *u*. This is obvious already from the fact that the initial magnetisation is positive.

Does the time evolution bring the system closer to the maximum entropy description in some sense, and how? The answer to these questions is the focus of the presented analysis and discussion.

### 3.1 Time Evolution of the Magnetisation

*t*, and just a copying of the spin state for the quiescent states, \(s_{j,t}\), at \(j \in \mathbb {Z}\), so that at time \(t+1\) we get

### Proposition 1

*i*. A local updating state \({\underline{s}}_{i,t}\) at position

*i*and time

*t*thus depends on \(2t-1\) initial stochastic variables, while a quiescent state \(s_{j,t}\) depends on \(2t+1\) initial stochastic variables.

### Proof

We prove this by induction. At time \(t=1\) the Q2R rule, Eqs. (2, 3), results in \({\underline{s}}_{i,1} = \xi _i\), and \(s_{j,1} = \xi _{j-1}+\xi _{j}+\xi _{j+1}\) (modulo 2). (The addition should here be understood as operating on the spin states, 0 and 1.) Thus Eqs. (4, 5) hold for \(t=1\).

*t*, then we can use the Q2R rule, Eqs. (2, 3), to find the expression for the states at time \(t+1\) (where all summations are assumed to be modulo 2),

*n*independent symbols (0 or 1) with probability

*p*for each 1, i.e.,

*t*is \({\sim }2t\). An odd number of 1’s in the sequence at \(t=0\) results in a 1 at the position at time

*t*. As

*t*increases, the probability for spin up, \(p(\uparrow ,t) = p(\underline{1},t)+p(1,t) \rightarrow 1/2\). Since the rule transforms an ergodic stochastic process description of configurations at time

*t*to a unique new such process at time \(t+1\), this implies that the magnetisation approaches 0. We define the magnetisation by the difference in spin probabilities, i.e., the average of upward spins (with direction \(+1\)) and downward spins (with direction \(-1\)),

### Proposition 2

*m*(

*t*), is given by

### Proof

*t*is given by two binomial distributions (for updating and quiescent states, respectively) and their corresponding probabilities for having an odd number of 1’s,

Thus we have an exponential convergence towards zero magnetisation. The frequency of spin up (and down) quickly approaches 1/2. For example, with an initial frequency of \(p=0.2\), we have after \(t=10\) time steps, \(p(\uparrow ,t)=0.499979...\) .

### 3.2 Time Evolution of Information-Theoretic Characteristics

In order to analyse how the Q2R dynamics transform the initial state (distribution of states) to states that in some way resemble equilibrium states, we make an information-theoretic analysis of the spatial configurations at the different time steps *t*, i.e., the stochastic processes that characterise those configurations.

#### 3.2.1 Information Theory for Symbol Sequences

*m*-length sequences at time

*t*, determined by the corresponding stochastic process characterising the spatial configuration, \(P_m(t) = \{p(x_1,...,x_m)\}_{x_i \in \{0,1,\underline{0},\underline{1} \}}\). All the key quantities for characterising order and disorder can be expressed in terms of block entropies, \(H_m(t)\),

*t*, is conserved since the dynamics is reversible. The conditional entropy

*h*can thus be expressed as

*m*increases quantifies correlation information \(k_m(t)\) in blocks over length

*m*,

*m*th symbol, on average, given that one knows already \(x_2,\ldots ,x_{m-1}\). With \(n=4\) possible states per lattice site, the total information of \(\log n=\log 4\) can be fully decomposed into the introduced quantities,

*m*, can be written

*h*is conserved in the time evolution, so is also the correlation information \(k_\text {corr}\). But the lengths at which correlation information is located may change over time, and thus we would in general expect the excess entropy, or the average correlation length, to change over time.

#### 3.2.2 Some Special Properties for the Q2R Model in One Dimension

*n*independently generated 0’s or 1’s (with probability

*p*for 1), i.e.,

#### 3.2.3 Correlation Characteristics of the Q2R Model in One Dimension

The following propositions assume that the initial state at time \(t=0\) is generated as described above: The spin states are generated by a Bernoulli process with probability *p* for spin up. Then the alternating order of updating and quiescent states are added on top of this. The entropy density at any *t* is then determined by the initial entropy density, i.e., \(h=-p \log p - (1-p) \log (1-p)\). The goal is to derive expressions that describe how the different contributions to the correlation information may change over time, and how that affects the estimates of the entropy density \(h_m(t)\).

### Proposition 3

### Proof

*Observation:*

*t*, the conditional probability of a quiescent state \(s_{m,t}\) does not change when adding new information in states beyond (to the left of) \(\underline{s}_{m-1,t}\). From Eqs. (4, 5) we see that \(s_{m,t}=\xi _{m-t}+...+\xi _{m+t}\) (mod 2), while \(\underline{s}_{m-1,t}=\xi _{m-t}+...+\xi _{m+t-2}\) (mod 2), i.e.,

Thus we know that, for \(m\ge 2\), \(k_{2m}^{(\times )}(t)=0\). From Eqs. (28, 29, 30) we see that \(k_{2m}^{(-)}(t)=k_{2m}^{(\times )}(t)=0\), and the Proposition then follows from Eq. (26). \(\square \)

### Proposition 4

### Proof

*m*, \(p(\underline{s}_1,...,\underline{s}_{m-1}, s_m) = p(\underline{s}_1,...,\underline{s}_{m-1}) p(s_m | \underline{s}_1,...,\underline{s}_{m-1})\). From the previous observation, Eq. (32), we find that \(p(\underline{s}_1,...,\underline{s}_{m-1}, s_m) = p(\underline{s}_1,...,\underline{s}_{m-1})\, p(\underline{s}_{m-1},s_m)\, / p(\underline{s}_{m-1})\), and this results in the entropy above. The same argument goes for odd

*m*.

*t*to a corresponding \((2m+1)\)-length sequence at time \(t-1\): \((s_1, \underline{s}_2',...,\underline{s}_{2m}', s_{2m+1})\), but with the opposite arrangement of updating and quiescent states. The updating states at time \(t-1\) may of course have different spin states. The one-to-one mapping, though, results in that the corresponding block entropies are the same, i.e., \(H_{2m+1}^{(-)}=H_{2m+1}'^{(\times )}\). The correlation information \(k_{2m+1}^{(-)}(t)\) can then be expressed in terms of block entropies at \(t-1\),

### Lemma 1

### Proof

The single site distribution is given by the \(\{1/2, 1/2\}\) distribution for updating and quiescent states multiplied with the distribution giving the probability for spin state 0 or 1 for the updating and quiescent states, \(P^{(2t-1)}=\{f_\text {odd}^{(2t-1)}(p), 1-f_\text {odd}^{(2t-1)}(p) \}\) and \(P^{(2t+1)}=\{f_\text {odd}^{(2t+1)}(p), 1-f_\text {odd}^{(2t+1)}(p) \}\) , respectively, in the case when \(t>0\). For \(t=0\), we instead have \(P^{(1)}=\{p,1-p\}\) for the spin state. This directly results in the Lemma. \(\square \)

### Lemma 2

### Proof

*h*from the Bernoulli process and the \(\log 2\) from the periodic structure, i.e., the two updating/quiescent possibilities \((-, \times )\) and \((\times , -)\) for pairs.

*t*, by \(p_t(\underline{s}_{i-1}, s_i) = p_t(s_i | \underline{s}_{i-1})\;p_t(\underline{s}_{i-1})\). By symmetry, the other order, \((s_{i-1}, \underline{s}_i)\), gives the same contribution to the block entropy. The resulting entropy can then be expressed as the sum of a conditional entropy and a single cell entropy,

### Proposition 5

### Proof

For \(t=0\), we only have correlation from the alternating updating and quiescent cells structure, i.e., \(\log 2\). For \(t>0\) we use the definition, \(k_2 = -H_2 + 2H_1\). Lemmas 1 and 2 then immediately result in the Proposition. \(\square \)

### Proposition 6

### Proof

For \(t=0\), there is obviously no correlation over blocks larger than 2.

*h*is). Propositions 3 and 4 imply that all correlation information from blocks of length

*m*, with \(m \ge 3\), is transferred to longer distances, \(m+2\), in the next time step. Therefore, the correlation of blocks over length 3, \(k_3(t)\), must come from the change in \(k_1 + k_2\) in the last time step, so that \(k_\text {corr}\) is conserved,

We note that Propositions 3–6 imply that the state at time *t* is characterized by a maximum information-theoretic correlation length of \(2t+1\), and thus a Markov process of memory 2*t*. Since we now have closed form expressions for all correlation information contributions, we can derive a closed form expression also for the excess entropy.

### Proposition 7

### Proof

We use the form of \(\eta \) which is a weighted sum over correlation information contributions, Eq. (24), i.e., \(\eta =\sum _m (m-1)k_m\). For \(t=0\) we only have contribution from \(k_2(0)=\log 2\), which gives \(\eta (0)=\log 2\).

This result shows that we have an average correlation length (in the information-theoretic sense), cf. Eq. (24), that increases linearly in time. The immediate implication is that some of the information that was initially detectable by looking at statistics over subsequences of shorter lengths is not any longer found at those length scales. How this relates to the approach towards a distribution closer to an equilibrium one is discussed in the following section.

## 4 The Microscopically Reversible Approach Towards the Equilibrium Distribution

The equilibrium distribution for a one-dimensional system can be derived by finding the distributions \(P_m\), over *m*-length blocks, that maximise the entropy density *h*, Eq. (15), under an energy constraint. Here the initial density *p* of spin up implies an energy density per site \(u=-(1-2p)^2\).

Because of the dual relation between entropy density *h* and correlation information \(k_\text {corr}\), as is seen in Eq. (21), the equilibrium can also be determined by minimisation of \(k_m\). With nearest neighbour interaction only, we can always choose \(k_m=0\) for \(m>2\), since the constraint need not give rise to higher order correlations [8]. As is well known, this results in magnetisation 0, or \(p(\uparrow )=p(\downarrow )=1/2\). With a normalization constraint this implies that \(p(\uparrow \uparrow )=p(\downarrow \downarrow )=1/2(1-2p(\uparrow \downarrow ))\), which with the energy constraint fully determines the distribution over pairs of spins. The solution is simply that \(p(\uparrow \downarrow )=p(1-p)\), i.e., the same as the initial one determined by the Bernoulli process. This follows from the fact that if energy is to be conserved, then \(p(\uparrow \uparrow )+p(\downarrow \downarrow )\) as well as \(p(\uparrow \downarrow )\) must be conserved.

### 4.1 Approaching the Equilibrium Characteristics

*m*-length block statistics, also that would converge towards \(h_\text {eq}\). This follows from the following observations.

*m*, the entropy density estimate, \(h_m(t)\), converges to the equilibrium entropy density value, \(h_\text {eq}\),

In Fig. 1, the entropy density estimate, \(h_m(t)\), as a function of the block length *m*, is depicted for the first four time steps as well as for time steps \(t=10\) and \(t=11\). Here it is clear that, for the first time steps, we will be able to detect the correct entropy density of the system, but as time proceeds the finite length estimates \(h_m(t)\) will converge towards a larger one, i.e., the equilibrium entropy density as we have shown above in Eq. (64).

*m*, is shown for the same time steps

*t*as above in Fig. 2. We note that, even though correlation information in total is conserved, a certain part of it is transferred to longer and longer blocks, as is stated by Proposition 4. Only one contribution remains at short length scales, \(k_2(t)\), which is determined by the 0 magnetisation and the energy constraint. The figure clearly illustrates that all correlation information, except \(k_2(t)\), will eventually become undetectable if finite block statistics is used which explains why finite length estimates of \(h_m(t)\) converges to \(h_\text {eq}\).

## 5 Discussion

In this paper we have investigated how, and in what sense, a microscopically reversible process can bring a system towards equilibrium. We have considered spatial configurations as generated by stationary ergodic stochastic processes, which has allowed us to study the characteristics of infinite systems directly from the beginning. Moreover, due to the ergodicity, a single microstate can be considered as a typical representative of the whole ensemble. We note that the reversible dynamics, given by the Q2R rule, implies that the (spatial) entropy density is conserved. We assume an initial state of independent spins generated by a Bernoulli process and non-zero magnetisation, i.e., \(p(\uparrow ,t=0) \ne \tfrac{1}{2}\). We have shown that under the reversible Q2R dynamics the system converges exponentially towards zero magnetisation — the equilibrium value.

*t*and \(m \ll 2t+1\) and, from Eq. (64), this can be expressed formally as

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