The Correlation Production in Thermodynamics

Macroscopic many-body systems always exhibit irreversible behaviors. However, in principle, the underlying microscopic dynamics of the many-body system, either the (quantum) von Neumann or (classical) Liouville equation, guarantees that the entropy of an isolated system does not change with time, which is quite confusing compared with the macroscopic irreversibility. We notice that indeed the macroscopic entropy increase in standard thermodynamics is associated with the correlation production inside the full ensemble state of the whole system. In open systems, the irreversible entropy production of the open system can be proved to be equivalent with the correlation production between the open system and its environment. During the free diffusion of an isolated ideal gas, the correlation between the spatial and momentum distributions is increasing monotonically, and it could well reproduce the entropy increase result in standard thermodynamics. In the presence of particle collisions, the single-particle distribution always approaches the Maxwell-Boltzmann distribution as its steady state, and its entropy increase indeed indicates the correlation production between the particles. In all these examples, the total entropy of the whole isolated system keeps constant, while the correlation production reproduces the irreversible entropy increase in the standard macroscopic thermodynamics. In this sense, the macroscopic irreversibility and the microscopic reversibility no longer contradict with each other.


I. INTRODUCTION
Considering a box of ideal gas with N particles, initially the particles occupy only part of the box, and then start to diffuse to the rest of the space (Fig. 1).After long enough time relaxation, they spread all over the box uniformly.Treating the initial and final states as in equilibrium, the entropy increase during this process is given by ∆S = N k B ln(V /V 0 ), where V (V 0 ) is the final (initial) occupied volume [1,2].This is a well-known example to demonstrate the macroscopic irreversibility and entropy increase in thermodynamics.But it is also known that the microstate probability distribution function (PDF) of this N -particle system evolves according to the Liouville equation, which obeys the time-reversal symmetry [1,[3][4][5], ∂ t ρ( P , Q, t) + {ρ( P , Q, t), H} = 0. ( Here { , } is the Poisson bracket, and ρ( P , Q, t) is the probability density around the microstate ( P , Q) := (p x 1 , p y 1 , p z 1 , . . .; q x 1 , q y 1 , q z 1 , . . . ) at time t.As a result, the Gibbs entropy of the whole system keeps a constant and never changes with time, This constant entropy seems in contradiction with our intuition of the "irreversibility" happening in the macroscopic world, like the above example of free expansion [4][5][6][7][8][9].Moreover, even if there are complicated particle-particle interactions, as long as H is a conservative Hamiltonian with no explicit time dependence, the Liouville dynamics still guarantees constant entropy, although in this case the system dynamics could be highly chaotic and unpredictable.
On the other hand, if there is no particle-particle interaction, the microstate evolution is well predictable, but the above "ir- reversible" diffusion process still happens in the box 1 .All these facts make the paradox more puzzling, and question our understandings of thermodynamics: how could the irreversibility and entropy increase arise from the microscopic dynamics obeying time-reversal symmetry [10]?
With this problem in mind, we study the full phase space dynamics of the ideal gas free expansion process.For ideal gas, the dynamics of each degree of freedom (DoF) is independent from each other, thus the problem can be simplified as studying the microstate PDF of only one DoF (p, x, t), and the exact analytical result is obtained [4,5].
For an equilibrium state, the microstate PDF must have a product form eq (p, x) = Λ(p) × Π(x), with Λ(p) as the Maxwell-Boltzmann (MB) distribution, and Π(x) = 1/L as the uniform spatial distribution.However, our result shows that once the diffusion starts, (p, x, t) can no longer be written in such a product form any more, that means, indeed it is NOT evolving towards the new equilibrium state as expected in the macroscopic intuition [4,5].
On the other hand, note that in practical experiments, usually it is the spatial and momentum distributions [P x (x) and P p (p)], but not the full phase space PDF (p, x), that are directly measured.We can prove P x (x, t) does approach the new uniform distribution over the whole space as its Figure 2. Demonstration for how the results are constructed for (a) periodic and (b) reflecting boundary conditions.The free space is cut into intervals of length L, and each contributes an "image" source.
steady state, no matter what initial state it starts from.In this sense, the above "irreversible" diffusion appears into our sight.Further, if particle collisions are considered, the singleparticle PDF, as the marginal distribution of the full N -particle ρ( P , Q), approaches an equilibrium state irreversibly.Therefore, although the total entropy does not change with time, the "correlation entropy" (mutual information), which quantifies the correlations between the marginal distributions (position-momentum and particle-particle), indeed better characterizes the irreversible entropy increase in thermodynamics [11][12][13].In this sense, the macroscopic irreversibility and the microscopic Liouville dynamics no longer contradict with each other.This idea also coincides with the recent correlation understanding of the irreversible entropy production in open systems [11,12,[14][15][16][17][18][19][20][21].

II. LIOUVILLE EQUATION FOR IDEAL GAS
Now we calculate the phase space dynamics of the above free expansion process.Since there is no interaction between particles, the dynamics of the 3N DoF are independent from each other, thus the total N -particle microstate PDF can be written as a product form ρ( P , Q, t) = i,σ (p σ i , q σ i , t) (σ = x, y, z) 2 , and we only need to study the microstate PDF (p, x, t) of a single DoF.For this one-body problem, the Liouville equation is with H = p 2 /2m.We can verify that any function with the form Φ[p, x − p m t] satisfies this differential relation.Now we further consider the initial and boundary conditions.
We assume initially the system starts from an equilibrium state confined in the area x ∈ [a, b] 2 Assuming there is no initial correlation between different DoF. Here 2 k B T as the average kinetic energy, and Z = √ 2π pT .Π(x) is the initial spatial distribution [Fig.3(a)] Such a product form indicates the spatial and momentum distributions are independent from each other and have no correlations in priori.
In the free space x ∈ (−∞, ∞), we can verify is the solution for the Liouville equation ( 3) with initial condition (4).For a confined area x ∈ [0, L] with periodic boundary condition (p, 0, t) = (p, L, t), the solution can be constructed with the help of the above free space one: Here F (p, x + nL, t) can be regarded as the contribution from the periodic "image" source in the interval The solutions (6,7) indicate that, once the evolution starts, the separable form (p, x, t) = f x (x, t) × f p (p, t), as the necessary condition for the equilibrium state, indeed does not hold any more.
In Fig. 3 we show the microstate PDF (p, x, t) at different times.As the time increases, the "stripe" in Fig. 3(a) becomes more and more inclined; once exceeding the boundary, it winds back from the other side due to the periodic boundary condition and generates a new "stripe".After very long time, more and more stripes appear, much denser and thinner, but would never occupy the whole phase space continuously [Fig.3(d, e)].
Fig. 3(e) shows the momentum distribution conditioned at the position x = L/2 taken from Fig. 3(d).In the limit t ∞, it approaches to an exotic function discontinuous everywhere, but not the MB distribution.
All the above features reveal that the microstate PDF (p, x, t) is indeed NOT evolving towards the new equilibrium state as expected from the macroscopic intuition, even in the long time limit.

III. MARGINAL DISTRIBUTIONS
It is also worth noticing that the spatial distribution P x (x, t) does approach the uniform one P x (x, t) 1/L as its steady state [Fig.3(d)].Now we show that indeed this is true for any initial state of Π(x) [4].
We first consider the initial spatial distribution is a highly concentrated one Π(x) = δ(x − x 0 ).From the analytical results (6, 7) of the microstate PDF (p, x, t), the spatial distribution P x (x, t) emerges as its marginal average: As demonstrated in Fig. 2(a), with the increase of time t, these Gaussian terms becomes wider and lower.Therefore, when t ∞, the spatial distribution P x (x, t) always approaches the uniform distribution in x ∈ [0, L].
Any initial spatial distribution can be regarded as certain combination of δ-functions, i.e., Π(x) = dx 0 Π(x 0 )δ(x − x 0 ).Therefore, for any initial Π(x), the spatial distribution P x (x, t) always approaches the uniform one as its steady state; at the steady state, there is no way to trace back what initial state it starts from.In this sense, although the underlying Liouville dynamics obeys the time-reversal symmetry, the "irreversible" diffusion appears into our sight.
On the other hand, P p (p, t) never changes with time, and always maintains its initial distribution, which can be proved by simple integral transforms: In the above discussions we assumed initially Λ(p) is the MB distribution in priori; if not, P p (p, t) would never be.

IV. REFLECTING BOUNDARY CONDITION
For the reflecting boundary condition, (p, x 0 , t + dt) = (−p, x 0 , t) (x 0 = 0, L), the analytical result can be also obtained by summing up the "reflection images" [Fig.2(b)], i.e., Here  To understand this, consider a particle moving in x ∈ [0, L], initially described by ˜ (0) (p, x, t); when it exceeds the boundary, ˜ (0) (p, x, t) no longer applies, then it is taken over by the next order ˜ (±1) , which is the reflection of ˜ (0) , and so on (Fig. 4).As a result, based on the same reason as above, the spatial distribution P x (x, t) also approaches the new uniform one as its steady state [4].
Under the reflecting boundary condition, the kinetic energy p 2 is still conserved, but the momentum p is not, thus P p (p, t) now varies with time.The reflection "transfers" the probability of momentum p to the area of −p.Therefore, in Fig. 4 we see that some "areas" of P p (p, t) are "cut" off from the initial MB distribution, and "added" to its mirror position along p = 0.If initially the spatial area [a, b] is in the center of [0, L], the probability changes from left-and right-reflection would cancel each other symmetrically, and P p (p, t) would keep the initial MB distribution, otherwise it would be changed by the reflections.
Since the reflection transfers the probability of p to its mirror position −p, the PDF difference δP p (t) := P p (t) − P p (0) is always an odd function [lower blue in Fig. 4(e)].As a result, the even moments p 2n of P p (t) are the same with the MB distribution, but the odd ones p 2n+1 are changed.
As the time increases, more and more "stripes" appear in P p (p, t), much thinner and denser.As a result, when calculating the odd orders p 2n+1 , the contributions from the nearest two stripes in δP p (t), positive and negative, tends to cancel each other.Therefore, in the limit t ∞, the odd orders p 2n+1 also approach the same value of the original MB distribution (zero) [Fig.5(c)].Namely, P p (p, t ∞) approaches an exotic function discontinuous everywhere, but all of its moments p n have the same values as the initial MB distribution.We emphasize these two distributions are different [23], and have different entropy, but in practical measurements it is difficult to tell their difference [5].
Again, here we see (p, x, t) is indeed not evolving towards the equilibrium state, but the concerning of partial information gives rise to the appearance of "irreversibility" [5].

V. CORRELATION ENTROPY
In practical experiments, usually it is not the full joint distribution (p, x) that is directly measured, but the single-particle PDF P x (x) and P p (p), which are given by the measurements of particle density and gas pressure respectively.
For periodic boundary case, P p (p) does not change, and P x (x) 1/L after long time, thus the above entropy increase (12) gives ∆ i S = ln(L/L 0 ), where L 0 := b − a is the length of the initially occupied area.Therefore, the full , which exactly reproduces the thermodynamics result in the very beginning [Fig.5(a)].
For reflecting boundary case, the spatial PDF P x (x) 1/L still holds and gives ∆S x = ln(L/L 0 ), but now P p (p) varies with time.Since the average energy p 2 is conserved, the initial thermal distribution should have the maximum entropy [24,27], thus the deviation of P p (p, t) from the initial MB distribution leads to a decrease of its entropy S p [P p ].The total change of the correlation entropy, ∆ i S = ∆S x + ∆S p , still increases monotonically [Fig.5(b)].
In this case, the correlation entropy exhibits the "irreversible" increase, yet does not exactly reproduce the thermodynamic result in the beginning.By further considering the particle collisions, this deviation can be healed, as studied in the Boltzmann H-theorem.

VI. BOLTZMANN H-THEOREM VS "ENTROPY DECREASE"
Notice that, in the study of the Boltzmann H-theorem, only the single-particle distribution f (p, r, t) is concerned, but not the total one ρ( P , Q, t) in the 6N -dimensional phase space [1,6,13].Thus f (p, r, t) also can be regarded as a marginal distribution that ignores the correlations between particles [28].
With no external forces, the Boltzmann equation is [1,29] This is just the above Liouville equation ( 3) plus an collision term (Appendix A).According to the Boltzmann Htheorem, the single-particle PDF f (p, r, t) always approaches the MB distribution as its steady state for any initial state [1,9,27].That means, when there are collisions between particles, f (p, r, t), as the single-particle marginal distribution of ρ( P , Q, t), also exhibits the behavior of irreversibility as well, and the above deviation from the MB distribution in the ideal gas can be eliminated.
As above, based on the inferred microstate PDF , which well reproduces the thermodynamic result in the beginning.Here ∆ i S characterizes the particleparticle and momentum-position correlations, and applies for both equilibrium and non-equilibrium states.
It turns out that the correlation entropy, rather than the total entropy, coincides closer to the irreversible entropy increase in thermodynamics.Now we show it could be possible, although not quite practical, to construct an "entropy decrease" process.
During this process, the total Gibbs entropy, which contains the full information, still keeps constant; the change of the correlation entropy ∆ i S [Eq.( 12)] would exactly experience the reversed "backward" evolution of Fig. 5, which is an "entropy decrease" process.This is just the idea of the Loschmidt's paradox [9,[29][30][31]], yet here is about the ideal gas with no particle collision.However, notice that such an initial state (0) is NOT an equilibrium state [6], and must be precisely prepared to contain very specific correlations of the marginal distributions [Fig.4(d)], thus the preparation is definitely non-trivial.Therefore, such an "entropy decrease" process is rarely seen in practice (except some special cases like the Hahn echo [32] and backpropagating wave [33,34]).

VII. SUMMARY
From the full phase space dynamics of the ideal gas expansion, it turns out that the ideal gas is indeed not evolving towards the new equilibrium state as expected from the macroscopic intuition.But if the partial information of marginal distributions is the only concerned or accessible measurement, the irreversibility appears to our sight, emerging from the ignorance of the correlations.The total entropy, which contains the full information, always keeps constant under the Liouville dynamics, but the "correlation entropy" characterizes the irreversible entropy increase in the macroscopic thermodynamics.In this sense, the macroscopic irreversibility and the Liouville dynamics with time-reversal symmetry no longer contradict with each other, and the "coarse-graining" was not needed through out the study [3,4,9].Besides the isolated classical system in this paper, this idea also coincides with the recent correlation understanding of the irreversible entropy production in open systems [11,12,[14][15][16][17][18][19][20][21], and may also be applied in more cases [35,36].time-reversal counterparts, thus equal to each other, and that gives Now we adopt "the assumption of molecular-disorder" [37], i.e., the two-particle joint probability can be approximately factorized as F (p 1 r 1 ; p 2 r 2 , t) f (p 1 , r 1 , t) × f (p 2 , r 2 , t).This is requiring that the correlation between these two particles is negligibly small.Then we obtain the Boltzmann transport equation, H-theorem -The Boltzmann H-function is defined as H := dς 1 f 1 ln f 1 , and its time derivative gives Replacing ∂ t f 1 with Eq. (A5), the diffusion term gives which can be turned into a surface integral and vanishes.And the collision term gives Exchanging the integral variables 1 ↔ 2 should give the same value, except the above ln f 1 is changed to be ln f 2 , and their combination gives Now exchanging the variables 12 ↔ 1 2 also keeps the same value, and their combination gives Here the transition ratio χ [12 1 2 ] is non-negative, and ≤ 0 always holds for any f α .Therefore, we obtain dH/dt ≤ 0, which means the function H(t) decreases monotonically.
Steady state -In the above inequality, the equality holds if and only if f 1 f 2 = f 1 f 2 , which means the collision induced increase ∆ (+) and decrease ∆ (−) of f (p, r) must balance each other everywhere, and that gives the steady state.
The logarithm of this equality reads ln f (p 1 , r 1 ) + ln f (p 2 , r 2 ) = ln f (p 1 , r 1 ) + ln f (p 2 , r 2 ), which depends on different variables on the two sides, and has a conservation form.Therefore, ln f must be a combination of these conservative quantities.During the collision (p 1 r 1 ; p 2 r 2 ) ↔ (p 1 r 1 ; p 2 r 2 ), the particles collides at the same position, and the total momentum and energy are conserved, thus we have ln f = C 0 + C i p i + C 4 p 2 .Therefore, f (p, r) is a Gaussian distribution of p at any position r.
Further, the diffusion term in Eq. (A5) requires p • ∇ r f = 0 in the steady state, thus f (p, r) must be homogenous for any position r.Therefore, the steady state of the Boltzmann equation (A5) is the MB distribution.
Molecular-disorder assumption -No doubt to say, the assumption of molecular disorder, F 12 f 1 × f 2 , takes an crucial role in the above discussion.Once two particles collide with each other, they get correlated.In a dilute gas, indeed collisions do not happen very frequently; if two particles collides with each other, they could hardly meet each other again.Therefore, if we assume initially there is no correlations between particles, we can expect that, on average, the collision induced inter-particle correlations are negligibly small, and thus this approximation holds well.
However, when we consider the "backward" evolution process as mentioned in the main text, as well as the Loschmidt paradox, the new "initial state", which is obtained from the "forward" evolution after some time, indeed has already established significant correlations between different particles during the forward evolution [28].This is similar with our discussion about the momentum-position correlation in the ideal gas expansion process.Therefore the approximation F 12 f 1 × f 2 does not apply in this case, and neither does the above Boltzmann equation (A5) and the H-theorem.

Figure 1 .
Figure 1.Demonstration for the ideal gas expansion in a box.

Figure 3 .
Figure 3. (Color online) (a-d) The distribution (p, x, t) in phase space at different times (τL := mL/pT as the time unit).As the time increases, Pp(p) does not change, but Px(x, t) approaches the new uniform distribution in x ∈ [0, L].(e) The conditional distribution at a fixed position (p, x = L/2) [vertical dashed line in (d)].
and R a [f (p, x)] := f (p, 2a − x) means making a mirror reflection to the function f (p, x) along the axis x = a.

t 1 LFigure 4 .
Figure 4. (Color online) (a-d) (p, x, t) under reflecting boundary condition, as well as its spatial and momentum distributions.(e) The momentum distribution from (d), and its difference (lower blue) with the initial MB one (green dashed lines).

Figure 5 .
Figure 5.The increase of the correlation entropy ∆iS for the (a) periodic and (b) reflecting boundary cases.(c) The evolution of odd moments p n t under reflecting boundary condition (the values have been normalized for comparison).The unit τL := mL/pT is the time for a particle with average kinetic energy pT to pass L.