Locality and evolution to equilibrium


 Quantum statistics and non-locality are deeply rooted in quantum mechanics and go beyond our intuition reflected in classical physics. Quantum statistics can be derived using statistical methods for indistinguishable particles - particles of quantum mechanics. Violation of strong locality - colloquially called the ghostly action at a distance - is one of the most amazing properties of nature derived from quantum mechanics. An intriguing question is whether the non-local evolution of indistinguishable particles is needed to reach the equilibrium state given by quantum statistics. Motivated by the above and similar questions, we developed a simple framework that allows us to follow space-time evolution of assembly of particles. It is based on a discrete-time Markov chain on countable space for indistinguishable particles. We summarise well-known and introduced new constraints on the transition matrix that grant space-time symmetries, locality of particle-transport, strong locality, and equilibrium state. Then, within the framework, several important cases are considered. First, we show that the simplest transition matrix leads to equilibrium but violates particle transport and strong localities. Furthermore, we construct a simple matrix that leads to equilibrium obeying particle-transport locality and violating strong locality. This resembles the properties of quantum mechanics. Finally, we demonstrate that it is also possible to reach equilibrium by obeying both particle-transport and strong localities. Thus, within this framework, the violation of a strong locality is not needed to reach the equilibrium of indistinguishable particles. However, to obey strong locality, a complex structure of the transition matrix is needed. In addition, we comment on distinguishable particles and, in particular, show that their evolution seen by an observer blind to particle differences may look like the evolution of indistinguishable particles with the properties of quantum mechanics. We hope that this work may help to study the relation between symmetries, localities and the evolution to equilibrium for indistinguishable and distinguishable particles.

The indistinguishable nature of particles and non-locality are deeply rooted in the world 1 modelled by quantum mechanics. Concerning the former one, the spin-statistics theorem [1- 2 3] relates the intrinsic spin of an indistinguishable particle to the quantum statistics it obeys. 3 The integer spin particles follow the Bose-Einstein statistics, whereas the half-integer the Fermi- 4 Dirac one. The latter property, violation of strong locality [4], indicates some sort of faster-5 than-light influence between remote events. The strong locality principle embeds a weaker but 6 more intuitive property -matter, in general, conserved quantities cannot be transported faster An intriguing question is whether the dynamics leading to the equilibrium of indistinguish-13 able particles requires violation of strong locality. In this paper, motivated by this and similar 14 questions, we introduce a simple framework that addresses the relation between localities, 15 symmetries and equilibration. It is based on a discrete-time Markov chain on a finite set of 16 microstates for indistinguishable particles. Then, within the framework several important cases 17 are considered. 18 The paper is organized as follows. Section I introduces the Markov chain for indistinguishable 19 particles. It also presents requirements on the transition matrix needed to obey space-time 20 symmetries, particle-number transport and strong localities, as well as reach an equilibrium-  The simplest way to address the locality of evolution and approach to equilibrium in a 27 dynamic model is to consider a 1+1 dimensional discrete-time Markov chain with a conserved 28 number of particles. The model is introduced in this section.

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The space is assumed to be a set of V discrete cells arranged in a one-dimensional ring -the 30 periodic boundary conditions are assumed. All cells are equivalent. For illustration a sketch 31 for V = 6 is presented in Fig. 1 (b).

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The total number of indistinguishable particles N is constant. For simplicity, only spin 33 zero particles are assumed. Thus, a microstate of the system is defined entirely by particle 34 multiplicities in cells, (n 1 , n 2 , .., n V ). Thus the total number of microstates (see e.g. Ref. [7]) is The system's evolution in time t is assumed to be discrete. Thus the evolution is given 36 by sequential transitions between its microstates. The time steps at which microstates of the 37 system are realized are equally distant. Transition probability from a microstate X at t to a 38 microstate Y at t + 1 is assumed to depend only on the initial microstate X and it is independent of microstates preceding X in time and the time step t. The above 40 assumptions, together with the probability transition matrix (2)  That is, in the steady state frequency of a transition from X to Y ,P (X)B(X → Y |X) is 46 equal to the one of the reverse transition from Y to X,P (Y )B(Y → X|Y ). HereP (X) andP (Y ) denote probabilities to find the system in microstates X and Y , respectively.

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(ii) Space-translation symmetry. Given a space translation T , the transition probability from 49 a microstate X to a microstate Y is the same as the transition probability from the 50 microstate T (X) to the microstate T (Y ).

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(iii) Space-reversal symmetry. Given any cell in the system, a reflection R of cell multiplicities 52 with respect to the cell does not change transition probabilities. The transition probability 53 from X to Y is the same as R(X) to R(Y ).

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The Cell Model with the space-time symmetries obeyed by the transition matrix is referred 55 to as a Symmetric Cell Model. The equilibrium state is defined as a steady state of maximum entropy. Here we present the 58 two conditions required to reach the equilibrium state within a Symmetric Cell Model. These 59 are ergodicity and transition-matrix symmetry, as we will explain in this section.

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The steady state refers to an ensemble of probabilities for the microstates to appearP (X), 61 which is left invariant by the probability transition matrix. In other words In the absence of other conditions than those defining the accessible states, as we will 63 assume in this work, it corresponds to the ensemble in which all microstates appear with equal 64 probability,P (X) =P (Y ) for any X and Y .  Ergodicity is sufficient to ensure that the system asymptotically goes to a unique steady state.

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It is also grants that time averages converge to the steady-state ensemble average in the limit We refer to a Symmetric Cell Model as equilibrating, if it is ergodic and it has the equilibrium-77 steady state. Equivalently, its probability transition matrix is ergodic and symmetric.

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Here the conditions needed for the locality of the system evolution within the Cell Model are 80 introduced. These are the conditions for transport and strong locality. The transport locality 81 denotes the locality of particle-number transport. The strong locality encompasses the transport 82 one, and, in addition, it includes a more subtle locality of correlations between remote events.

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Transport locality. Particle trajectories and velocities are defined for distinguishable particles 84 and the transport locality requirement corresponds to the well-known requirement that particle

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In the case of indistinguishable particles, the maximum speed of particle-number transport, 90 coupled to the particle number conservation implies the following. During a single time-step 91 the particle number in any interval of cells cannot be transported beyond an interval by ∆ cells 92 longer on the left and right. And it cannot be squeezed to an interval by ∆ cells shorter on the 93 left and right. This provides two transport-locality conditions: where k = 0, 1, . . . and n X j , n Y l are particle numbers in cells j, l of initial (X) and final (Y ) 95 microstates, respectively.

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Strong locality. According to the strong locality principle, the probability of an event E at time 97 t + 1 can be influenced only by events within its past light cone. Thus E has to be independent 98 of events outside its light cone when possible correlations due to the common past are removed.

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Referring to Fig. 1, given an initial configuration at the time-step t, any event E at the time-100 step t + 1 can depend only on the configurations of the system at time t, within the same cell, 101 and in the ±∆ neighbours (the dark orange line in the plot (a)). The event E (for example 102 particle number in cell "2" at t + 1) is independent of all events θ in gray areas (cells) for all possible events F and θ. Note that events in the yellow regions can be correlated with 104 E because possible common-past correlations are not removed.

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Violation of strong locality can be quantified by introducing the measure: where P (E | (F, θ) ) is obtained by averaging P (E | (F, θ) ) over all possible θ for given E and F .

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Obviously, for an evolution obeying strong locality and thus V SL = 0.
where P(B k ) is a normalised to unity number of transition-matrix elements with B(X → (b) The matrix which leads to equilibrium obeying transport locality and violating strong locality.
(c): The matrix which leads to non-equilibrium steady state obeying transport and strong locality.
(d): The matrix which leads to equilibrium obeying transport and strong locality.
Microstates are ordered according to their sequential number S(X) defined as a position of the microstate in a vector of "human-friendly" microstates labels calculated as L(X) = V i=1 n V −i · 10 i−1 . The colour scale indicates values of transition probabilities, with the white colour denoting zero probability. Note different colour scales adopted to underline qualitative differences between the matrices.

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The simplest transition matrix is the one which has all elements equal: where W is the total number of microstates given by Eq. (1). The matrix has no free parameters, The simplest matrix leading to equilibrium and obeying transport locality is constructed as 136 follows. All off diagonal elements corresponding to transport-local transitions are equal and 137 non-zero. Elements for transitions violating transport locality (teleportation) are set to zero.

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Then diagonal elements are given by unitarity. This procedure leaves one free parameter.

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The matrix is ergodic; see Methods. By construction, the matrix is symmetric. Thus, it 140 corresponds to an equilibrating Symmetric Cell Model.

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By definition, the matrix obeys transport locality. However, it violates strong locality. This It can be shown that the transition matrix is ergodic. However, it is not symmetric despite 158 being time-reversible. The rather technical, analytic proof for that is presented in Methods.

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Thus, the steady state is not the equilibrium one.

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In summary, the transition matrix defined above is of the type [EQ− TL+ SL+]. An 161 example matrix of this type calculated for V = 6 and N = 4 is shown in Fig. 2 (bottom-left).

D. Transition-matrix type [EQ+ TL+ SL+]
164 Here, we show that it is possible to construct a transition matrix that leads to equilibrium 165 as well as obeys transport and strong locality. The construction utilises the idea introduced in  In summary, the transition matrix defined above is of the type [EQ+ TL+ SL+]. An 188 example matrix of this type calculated for V = 6 and N = 4 is shown in Fig. 2 (top-right). The Then, it is possible to prove that is a steady-state for the distinguishable particles, and it fulfills the time-reversal condition This, together with ergodicity, shows thatP (X) is the unique-steady state of the system.

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Using (12) and (14), one gets In the last equality we have used the time reversibility of the transition matrix for indistin-232 guishable particles. In order to complete the proof, one notes that The left-hand side in Eq. (17)

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Finally, one gets This is the time-reversibility condition for the distinguishable particle case. From the unitarity 241 of B(X → Y|X ) follows that P (X ) is a steady state.

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Thus we conclude that one can construct a transition matrix for distinguishable particles, 243 leading to the evolution that for an observer blind to differences between particles obeys space-244 time symmetries and transport or strong locality, and leads to equilibrium. In particular, 245 a matrix leading to an evolution compatible with quantum mechanics, like in Sec. II B and 246 Sec. II D, for the observer blind to particle differences can be obtained.

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Distinguishable particles and quantum mechanics. We note that, in general, that is, the transpose of the probability transition matrix is not normalised to one. This is, in 249 fact, enough to prove that it cannot stem from quantum dynamics. Before proving the latter, 250 we check that (19) is correct. The simplest way to do this is to select a final microstate Y 0 251 as a microstate with all particles in the same cell. Thus Y 0 is invariant under permutation of 252 particle. Therefore, independently on the initial state X , because of the definition (12), one for any X that can go to Y 0 , since, no matter the initial microstate, there is only one permutation of particles in the final one.

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By assumption, one has However, the sum over the X microstates repeats probabilities for each equivalence class X =

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[X ] a number of times equal to the number of permutations P(X), therefore where the inequality follows from the fact that (at least some of) the microstates of distinguish-for a generic HamiltonianĤ, and using the natural units. Assuming that δt = t i+1 − t i is Consequently, the evolution of distinguishable particles, which leads to the equilibrium- sition matrix with zero information entropy leads to the equilibrium-steady state by violating 282 particle transport and strong localities. Furthermore, we construct system evolution that leads 283 to an equilibrium-steady state obeying particle-transport locality and violating strong locality.

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This resembles the properties of quantum mechanics. Finally, we show that it is also possible 285 to reach equilibrium by obeying both particle-transport and strong localities. Thus, within 286 the framework, the violation of a strong locality is not required to reach the equilibrium of 287 indistinguishable particles. However, to obey strong locality, a high information entropy of the 288 transition matrix is needed.

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Additionally, we show that one can define a transition matrix for distinguishable parti-290 cles, leading to the evolution that for an observer blind to particle differences has "quantum-291 mechanical" properties -obeys space-time symmetries and particle-transport locality violates 292 strong locality and leads to equilibrium. 293 We hope that this work opens new possibilities for the study of the relation between space-294 time symmetries, particle-transport and strong localities and evolution to equilibrium for in-295 distinguishable and distinguishable particles. In particular, future studies may address the role 296 of space-time symmetries and include processes of particle creation and annihilation.