Classical and Quantum Models in Non-Equilibrium Statistical Mechanics: Moment Methods and Long-Time Approximations

A quite wide and very interesting set of references, from different standpoints, related to and/or oriented towards the foundations of non-equilibrium Statistical Mechanics can be seen in [1]. Non-equilibrium statistical systems of classical particles are described by Liouville classical distribution functions ($W_c$) [2,3,4,5]. For non-equilibrium statistical systems of quantum particles, several distribution functions are available, Wigner functions (W) [3,4,5,6,7] being a suitable and simple possibility in a global sense (in spite of the fact that their positivity is not warranted). Open statistical systems (involving, typically, one or a few degrees of freedom) are those subject to external “heat baths” ($hb$) or reservoirs at thermal equilibrium at temperature T, with finite or small dissipation due to the $hb$. We shall regard vanishing dissipation as a mathematical idealization of the case with small friction. Closed statistical systems are those not subject to any external $hb$, and they are formed by an enormously large number of interacting particles. General equations describing, in differential form, the time evolutions of non-equilibrium closed interacting classical or quantum systems (with some initial off-equilibrium state) are well known [2,3,4,5]. On the other hand, approximate time evolution differential equations for the effective or relevant degrees of freedom for non-equilibrium open or closed (classical or quantum) interacting systems are known in certain cases [2,3,4,6,7]. Generically, those evolution equations turn out to be increasingly difficult to solve (even on a computer), as the number of particles in the system grows and/or one tries to describe the state of the system for increasingly long times (starting from some initial non-equilibrium state). On the other hand, there does exist certain global understanding of various classes of solutions of those evolution equations, at least in a number of cases. Anyway, to achieve wider and better knowledge of how open or closed (classical or quantum) statistical interacting systems evolve in time continues to play a key role in Statistical Physics. In particular, an issue of crucial importance is how irreversibility could arise in the long-time evolution of closed classical or quantum many-particle systems [2,3,4,5,8], or stated in other words, how an arrow of time could set in.

Non-equilibrium open one-particle statistical classical systems, subject to potentials and to dissipation due to the $hb$, lend themselves typically to perform certain constructions and approximations in a framework which is simpler than those met for other cases: see [9,10,11,12] and references therein. We shall remind them [9,10,11,12] firstly, in order to motivate our developments. In them, classical time-independent equilibrium distributions ($W_{c,eq}$), which are independent on dissipation, have been used as weight functions to generate families of orthogonal polynomials in momenta. The latter has been possible due to two distinguishing and simplifying features of $W_{c,eq}$: dependences on classical momenta and those on spatial coordinates factorize, and $W_{c,eq}$ is Gaussian in classical momenta. Then, the orthogonal polynomials generated by $W_{c,eq}$ turned out to be just the standard Hermite polynomials [13] and, moreover, they are independent on spatial coordinates (and on dissipation). Those orthogonal polynomials yield (by integrating over classical momenta) moments $W_{c,n}$ of the non-equilibrium classical distributions $W_c$. The non-equilibrium $W_{c,n}$’s depend on spatial coordinates and time (t) and on dissipation (but not on momenta), and fulfill an infinite linear hierarchy. A key issue was that, for long times, the lowest moment appears to dominate the evolution towards thermal equilibrium with the $hb$ (and, hence, to become independent on the initial non-equilibrium state and on dissipation) [9,10].

Interesting open issues in [9,10,11,12], among others, include: (a) further study of the process leading to moments which depend only on spatial coordinates; (b) properties of the solutions of the non-equilibrium classical hierarchies for different potentials (vanishing or not at large distances) and their behaviour for long times; (c) generalizations to non-equilibrium closed classical interacting many-particle systems without external $hb$’s (with potentials vanishing at large distances), with initial states at thermal equilibrium at temperature T at large distances but off-equilibrium at finite distances and with the possibility of obtaining, in a suitable long-time approximation, irreversibility; (d) generalizations of (c) when classical electromagnetic degrees of freedom are included; (e) extensions to open one-particle quantum systems by means of Wigner functions (either with or without dissipation); (f) closed quantum interacting many-particle systems, which clearly lead to far more difficult problems. The above non-equilibrium moment method relies, very crucially, on the previous knowledge and properties of equilibrium distributions. Being natural to work with quantum Wigner functions (although not mandatory), we anticipate certain genuine quantum difficulties: the equilibrium quantum Wigner functions are neither Gaussian in momenta nor known in closed form, and dependences on momenta and those on spatial coordinates do not factorize, in general. The issue of the possible dependence of the quantum equilibrium distributions on dissipation leads to additional complications. The use of other quantum distribution functions does not appear to yield improvements in a global sense.

We shall review in outline items (a)–(c), analyzed in our previous work [14,15,16,17,18,19]. We shall study further convergence aspects for issues (b) and (c), which, even if shortly, will provide some additional partial clarification, so far unpublished. We shall present a simple overview of [14] on long-time approximations and irreversibility in closed interacting many-particle classical systems, including some improvements from [16] and [18,19]. Item (d) has been dealt with in [15] in detail and we shall not add more here. And, finally, we shall concentrate on (e). The genuine difficulties of quantum cases anticipated above already show up in one-dimensional models. Some generalizations of the moment method to non-equilibrium open one-particle quantum interacting systems, without or with dissipation and to lowest order in Planck’s constant, have already been studied [18,20]. Here, we shall concentrate on constructing the equilibrium Wigner functions, the families of orthogonal polynomials generated by the latter and non-equilibrium moments and equations at low order in the hierarchies, to all orders in Planck’s constant, in one-dimensional models. Models in which the equilibrium Wigner functions either depend on dissipation or are independent on it will be analyzed. Item (f) above (closed many-particle interacting quantum systems and irreversibility issues) will not be addressed in this work.

This paper is organized as follows. Section 2 reviews open one-dimensional classical systems with or without dissipation. Section 3 treats classical closed interacting many-particle systems. Section 4 deals with general aspects of quantum-mechanical one-dimensional models, in the idealized limit of vanishing dissipation. Section 5 is devoted to open quantum-mechanical one-dimensional models with quartic plus quadratic potential, also without dissipation, through an approach different from that in Section 4. Section 6 treats open quantum-mechanical one-dimensional models with quadratic plus quartic potential with dissipation. Various technical aspects in Section 4, Section 5 and Section 6 are treated in Appendices A–D. Finally, Section 7 presents some conclusions and various discussions. A short account of the contents of Section 5 and Section 6 has been presented orally in the 11th International Conference on Orthogonal Polynomials, Special Functions and Applications, held in Universidad Carlos III, Madrid, Spain (August 29 through September 2, 2011).

We shall present an outline of the main developments, omitting lengthy arguments and equations, which can be seen in [14]. We treat a closed large system of many ($N\gg 1$) classical nonrelativistic particles, in d spatial dimensions ($d=1,2,3$), with spatial coordinates ${\mathbf{x}}_1$,…,${\mathbf{x}}_N$ ($\equiv [x]$) and momenta ${\mathbf{q}}_1$,…,${\mathbf{q}}_N$ ($\equiv [q]$). All particles, which are identical, have mass m. Let $x_{i,\alpha }$ and $q_{i,\alpha }$ be the Cartesian components of ${\mathbf{x}}_i$ and ${\mathbf{q}}_i$, respectively ($i=1,\dots ,N$, $\alpha =1,\dots d$). Neither a $hb$ nor external friction mechanisms nor external forces are assumed. The interaction potential is: $V={\Sigma }_{i,j=1,i<j}^N{V}_{i,j}(\mid {\mathbf{x}}_i-{\mathbf{x}}_j\mid )$ and we suppose that all $V_{i,j}(\mid {\mathbf{x}}_i-{\mathbf{x}}_j\mid )$ are repulsive ($\ge 0$) and tend quickly to zero for large $\mid {\mathbf{x}}_i-{\mathbf{x}}_j\mid$. The physical idea is that the very large number of degrees of freedom (which be at thermal equilibrium with one another) in the system play the role of a $hb$. The non-equilibrium classical distribution function is: $W_c=W_c([x],[q];t)$. Boltzmann’s equilibrium (canonical) distribution at absolute temperature T is : $W_{c,eq}=exp[-\beta ({(2m)}^{-1}{\Sigma }_{i=1}^N{\Sigma }_{\alpha =1}^d{q}_{i,\alpha }^2+V)]$.

The initial non-equilibrium distribution $W_{c,in}$, to be regarded as known, is quite arbitrary in practice. Instead of considering the most general initial state, we shall choose a class of $W_{c,in}$’s, which will be: (i) qualitatively consistent with the idea [8,22], typical of Information Theory, that one should employ only distribution functions compatible with the limited information available which, in turn, refers to expectation values not of all possible dynamical variables but only of an observable subset of such variables (these ideas being imposed for $t=0$ only, but not for $t>0$); (ii) also consistent with standard variables employed in Equilibrium Statistical Mechanics and Fluid Dynamics [2,3,4,5]. Then, we shall treat a class of an explicit ansätze for $W_{c,in}$ which will depend on a finite number of functions (actually, $2+d$) of one single position vector, $\mathbf{x}$: ${\lambda }_k={\lambda }_k(\mathbf{x})$, $k=0,2$, and ${\lambda }_{1,\alpha }={\lambda }_{1,\alpha }(\mathbf{x})$, $\alpha =1,\dots ,d$ (all of them being independent on time and on momenta). The expression for $W_{c,in}$ in terms of ${\lambda }_k$ and ${\lambda }_{1,\alpha }$ has appeared previously [3,5] (and is related to the Massieu-Planck function [5]). In short, $W_{c,in}$ at $t=0$ will be chosen to describe thermal equilibrium with homogeneous temperature T for large distances but non-equilibrium for intermediate and short distances (with spatial inhomogeneities). The ansatz is: Consistently with (i)–(ii) above, the λ’s will be uniquely determined (through standard recipes in Information Theory) in terms of $2+d$$\mathbf{x}$-dependent observables (also independent on time and on momenta) typically employed in Fluid Dynamics, which, by assumption, are known at $t=0$: mass density, fluid velocity and some suitable energy density [5,14]. For simplicity, no observables associated to angular momentum are included. How does the equilibrium temperature T appear in this closed system, without $hb$? We assume that $W_{c,in}$ describes thermal equilibrium at temperature T at large $\mathbf{x}$ but off-equilibrium at finite $\mathbf{x}$. We accept that ${\lambda }_2(\mathbf{x})$ approaches quickly a non-vanishing constant, ${\lambda }_2(\infty )$, as $\mid \mathbf{x}\mid$ tends to ∞ along any direction and that the same holds for ${\lambda }_0(\mathbf{x})$. A similar statement holds for ${\lambda }_{1,\alpha }(\mathbf{x})$, the corresponding (large-$\mid \mathbf{x}\mid$) limiting value being zero. At finite $\mathbf{x}$, the off-equilibrium ${\lambda }_2(\mathbf{x})$, ${\lambda }_0(\mathbf{x})$ and ${\lambda }_{1,\alpha }(\mathbf{x})$ do depend on $\mathbf{x}$ and, so, differ from their respective constant (large-$\mid \mathbf{x}\mid$) limiting values, which describe equilibrium. Then, consistency is achieved (T being thereby introduced) if, in the thermodynamical limit, ${\lambda }_2(\infty )$ tends to ${(k_{B}T)}^{-1}$ (plus corrections which approach zero in that limit). The dominant contributions to various statistical averages at $t=0$ come from large $\mathbf{x}$, up to corrections which tend to vanish as N increases. The very large number of degrees of freedom involved in the largest part of the system (for large $\mid \mathbf{x}\mid$) are at equilibrium at T. For consistency, the $2+d$$\mathbf{x}$-dependent dynamical variables known at $t=0$ (mass density, fluid velocity and some suitable energy density) have to fulfill the corresponding behaviour as $\mid \mathbf{x}\mid$ tends to ∞ along any direction. For details, see [14].

The reversible Liouville equation reads: Let $[n]$ denote a set of nonnegative integers $(n(i=1,\alpha =1),\dots ,n(i=N,\alpha =d))$ and let $n={\Sigma }_{l=1}^N{\Sigma }_{\alpha =1}^{d}n(l,\alpha )$. Let $\left[dq\right]={\prod }_{i=1}^N{\prod }_{\alpha =1}^{d}d{q}_{i,\alpha }$. We introduce non-equilibrium moments $W_{[n]}$ of W (using products of Hermite polynomials, by generalizing Equation (3) with all integrations in ($-\infty ,+\infty$)): If $W_c=W_{c,eq}$, then $W_{c,eq}([0])$ ($[0]=(0,0,\dots ,0)=(n(i=1,\alpha =1)=0,\dots ,n(i=N,\alpha =d)=0)=[n=0]$) is proportional to $exp[-\beta V]$ and $W_{c,eq}([n])=0$, $[n]\ne [0]$ (say, $n\ne 0$). Equation (26) can also be applied to $W_{c,in}$ and gives the corresponding initial moments, $W_{c,in}([n])$. We shall work with the symmetrized moments $g([n])=W_{c,eq}{([0])}^{-1/2}{W}_c([n])$. One gets an infinite reversible three-term linear recurrence for $g([n])$’s, generalizing Equations (4) and (5). It reads: The Laplace transform of the hierarchy Equation (27) is the actual many-particle counterpart of Equation (6) with ${\sigma }^{-1}=0$. Such a Laplace transform (with $N\gg 1$) can be formally solved in terms of linear operators $D[[n];s]$, which generalize the previous $D[n;s]$. For details, see [14]. All $D[[n];s]$ are square matrices, due to the indices $i=1,\dots ,N$ and $\alpha =1,\dots ,d$. In turn, each matrix element in those square matrices is an ordinary linear integral operator, arising from the partial differential operators $M_{l,\alpha ;n(l,\alpha );+}$ and $M_{l,\alpha ;n(l,\alpha );-}$, as l, α and $n(l,\alpha )$ vary. The $D[[n];s]$ fulfill the following formal hierarchy [which generalizes Equation (7)]: The linear operators $M_{\pm ,[n]}$ are rectangular matrices, the elements of which are formed out of the partial differential operators $M_{l,\alpha ;n(l,\alpha );+}$ and $M_{l,\alpha ;n(l,\alpha );-}$. $M_{+,[n+1]}$ can be shown to be the adjoint of $-M_{-,[n]}$. By iterating Equation (30) indefinitely, one can express formally the linear operator $D[[n];s]$ as an operator continued fraction, which depends on all partial differential operators $M_{l,\alpha ;n(l,\alpha );+}$ and $M_{l,\alpha ;n(l,\alpha );-}$ and generalizes the operator continued fraction for $D[n;s]$. See [14]

By generalizing the iterative arguments in Subsection 2.3, it follows that, for both $V\ne 0$ and $V=0$, $D[[n];ϵ]$ for $ϵ>0$ is a Hermitian operator with non-negative eigenvalues for $n\ge n_0\ge 1$ [14].

A few remarks for the case $V\equiv 0$ may be clarifying. In the Laplace transform of Equation (27), let us perform a spatial Fourier transformation from configuration space (${\mathbf{x}}_1$,…, ${\mathbf{x}}_N$) to wavevector space (${\mathbf{k}}_1$,…, ${\mathbf{k}}_N$)$\equiv [k]$. Let $e(k;N)\equiv k_{B}T{\sum }_{j=1}^N{(2m)}^{-1}{\mathbf{k}}_j^2$. Then, the Fourier transform $D_1[[k];[n];s]$ of $D[[n];s]$ for Re$s>0$ is an ordinary continued fraction, given in Equations (16)–(18) with $e(k)$ replaced by $e(k;N)$. Then, with such a replacement, the properties of $D_1(k;n;s)$ given in Subsection 2.4 also hold for $D_1[[k];[n];s]$. Notice that $D_1[[k];[n];s=0]$ diverges as $e{(k;N)}^{-1/2}$ if $e(k;N)\to 0$. On the other hand, and contrary to what happened for one particle in one spatial dimension [recall the comment after Equation (21)], $\int \left[dk\right]D_1[[k];[n];s=0]$ converges near $e(k;N)=0$. This would suggest that the actual counterpart of Equation (9) (containing $D[[n];ϵ]$ acting on various $M_{l,\alpha ;n(l,\alpha );-}g(n(1,1),\dots ,n(l,\alpha )-1,\dots ,n(N,d))$’s), when integrated over $[k]$’s to come back to $[x]$-space, would converge at small $[k]$’s as $ϵ\to 0$, for the actual $V\ne 0$. Whether such a property holds is an open question.

In spite of the very involved structure of Equation (27) (and of its Laplace transform), we argue that a simple long-time approximation can be performed in it, for $V_{i,j}\ge 0$ (and vanishing quickly at large distances) and very large N (in the thermodynamical limit), which generalizes that in Subsection 2.3. This approximation consists in fixing $s=ϵ>0$ (ϵ being small) in the whole hierarchy of operators $D[[n];s]$, for any $n\ge n_0(>0)$ which, then, become Hermitian operators $D[[n];ϵ]$ with no negative eigenvalues. It is crucial that s-dependences be kept in $D[[n];s]$, for $n<n_0$. Notice that the non-vanishing factors $n{(l,\alpha )}^{1/2}$ and ${((n(l,\alpha )+1))}^{1/2}$ in $M_{l,\alpha ;n(l,\alpha );-}$ and $M_{l,\alpha ;n(l,\alpha );+}$, respectively, tend to reduce, as the $n(l,\alpha )$’s increase, the relative importance of having fixed $s=ϵ$ and the contribution of the latter in the $D[[n];s=ϵ]$’s with $n\ge n_0$. This is a genuine feature of the $D[[n];s=ϵ]$’s. See [14], where it was seen that, by imposing $n_0>2$, the long-time approximation is exactly consistent with all hydrodynamical balance equations. For simplicity, we discard all the initial moments $W_{c,in}([n])$ for $n\ge n_0$. We regard $D[[n_0];ϵ]$ as a fixed (s-independent) operator, yielding all $g([n_0])$ in terms of all $g([n_0-1])$. The choice $ϵ=0$ was made in [14]. By using [16] and [18,19], the arguments in [14] are easily seen to hold for $ϵ>0$. All that leads to a closed approximate hierarchy for $g([n])$’s (with initial moments $W_{c,in}([n])$), with $n<n_0$, which appears to yield an approximate irreversible evolution towards thermal equilibrium at T. $g([n])$’s and, then, $W_c([n])$ relax the quicker the larger n, provided that $n<n_0$. $W_c([0])$ would dominate the approach towards equilibrium for $t\to +\infty$. See [14]. All that appears to work based on the general properties of $D[[n_0];ϵ]$: for quantitative studies, some ansatz or approximation should be provided directly for it. An arrow of time would follow approximately, in the present case.