People use master equations (ME) to obtain equations of motion for a reduced density operator or for the probability distribution (PD) of a subsystem of interest S in interaction with (a usually much larger) heath bath. The key topic is that S , our system at hand, is in an out of equilibrium state. Master equations are standard tools in statistical physics and related areas. See for instance [1] and references therein. The core issue is how to gather relevant information on S from the concomitant out of equilibrium dynamics. Thus, our goal is to describe the time evolution of the PD for the system S . This should be done by recourse to a master equation complying with some basic requirements. First, the concomitant dynamics guarantees positivity and normalization of the time-dependent PD. Second, for a constant heat-bath temperature, the PD satisfies an appropriate H-theorem consistent with the PD’s relaxation towards equilibrium. Last, for a constant heat-bath temperature, the associated Gibbs’ distribution is a stationary solution of the master equation. The solution to this problem is found by recourse to an appropriate master equation (ME) [1]. A nice instantiation of the ME-procedure is presented by Takada, Conradt, and Richet (TCR) in [2], that we discuss below for didactic purposes.

Contemplate a two-levels (1 and 2) model (system S ) in contact with a heath bath B of temperature T. If the population P 2 of the excited state is calculated, then P 1 becomes automatically determined. The inter-level transition rate is the critical parameter ruling the degree of non-equilibrium. We call p 1 and p 2 the two occupation probabilities.

Now imagine a heating and cooling operation in which the reservoir’s temperature varies as with time in this manner: 1) T first grows, reaches a maximum value T M (arbitrary units (AU)), and then 2) diminishes, returning to its initial value T 0 = 1 (AU). TCR interpret this environment as describing a particle movable in an asymmetric double-well potential. Site 1 is the bottom of the first well (of energy is E 1 ). Analogously, site 2 is the bottom of a second well, of a higher energy E 2 . Thus, E 1 is a potential energy barrier between states 1 and 2 to be overcome. System S evolves after wards with an energy decrease to E 2 , leading to the second state. The temperature describes an isosceles time-triangle of apex T M and base 2 τ . Takada, Conradt, and Richet concoct the concomitant master equation (ME) as

d p 1 / d t = − a 1 p 1 + a 2 p 2 ;       p 1 + p 2 = 1, (1)

with

a i = exp [ − E i / k b T ] ,       i = 1 , 2 , (2)

where k B is Boltzmann’s constant, to be set equal to unity here from.

Inspired by [2], we will tackle a different problem. We focus attention on a quantum many-body system S of interacting fermions, proposed in [3], whose interaction is governed by an SU2 algebra. S is first heated and then cooled in the time-triangle fashion described above. We will deal with a master coupled system of N equations (not just two as in [2]). The heating-cooling operation competes with the fermion-fermion interaction. A Gibbs’ canonical ensemble treatment of the model advanced in [3] is detailed in [4]. At the same time, for each distinct temperature T we contemplate a fictional coupled system S -heat bath in thermal equilibrium at any extant T-value [this is called the quasi-static approximation (qsa)]. Of course, at that time t at which the solutions of our master equation coincide with those of the qsa we will have reached equilibrium.

Our main research tools will be suitable information theory quantifiers. With them we will assess “distances” between the “ME probability distribution” (MEPD) and the “quasi-static” PD. The ensuing numerical values will indicate just different the two treatments MEPD and qsa are. This yields a quantitative measure of how far on is from equilibrium. To objectively measure this is our central goal here. Epistemology teaches classification is an essential ingredient of Science [5]. Classification entails numbers, and the distances that we are seeking do supply such numbers in the present context.

Section 2 describes our master equation. Suitable details of our exactly solvable quantum many-body system, (the laboratory to test our thermal distances between off equilibrium and equilibrium) are the subject of Section 3. Section 4 discusses features about thermal quantifiers that will serve as “distances”. The main present results are displayed exhibited and talked through in Section 5 and, finally, some conclusions are elaborated in Section 6.

Introduce first an antisymmetric matrix A whose elements A i j have the following properties:

A i j = ∑ i , j = 1 N [ p j exp ( − β E i ) − p i exp ( − β E j ) ] , (6)

so that A i j = − A j i . Then,

∑ i = 1 N p i d t = ∑ i , j = 1 N   c i j A i j = ∑ i , j = 1 ;     i   l e j N   c i j ( A i j + A j i ) = 0 , (7)

q.e.d.

A much simpler ME was recently studied in Ref. [6]. Here we permit interaction only between neighbor levels (with one exception demanded by normalization). This entails some sort of selection rules that allow for only certain kinds of transitions. The simplified ME reads

d p k d t = p k − 1 exp ( − β E k ) − p k exp ( − β E k − 1 ) , (8)

and we continue in this manner till we encounter

d p 0 d t = − [ p n − 1 exp ( − β E n ) − p n exp ( − β E n − 1 ) ] − ⋯                   − [ p 0 exp ( − β E 1 ) − p 1 exp ( − β E 0 ) ] ,

where the last relation guarantees normalization. The initial conditions are p i ( 0 ) = exp ( − E i / k T ) / Z . for all i. β depends upon time in the peculiar triangular way described above. We set

β ( t ) = 1 / T ( t ) ,

together with

E j = j ;         all     j .

If β were constant, then the time-dependent probabilities p i ( t ) would relax to stationary Gibbs-distributions. We will compare below results, in particular equilibrium attainment, for this simpler ME with those yielded by the more complete ME described above.

As V increases from zero, the ugs energy E o is not affected at once. It keeps its value till a critical V-specific value is reached, of 1 / ( M − 1 ) . At this stage, the interacting gs instantaneously becomes | J , − M / 2 + 1 〉 . If V continues growing, additional new phase transitions (PT) occur. The PT between J z = − k and J z = − k + 1 tales place at V = 1 / ( 2 k − 1 ) . The successive PT’s end up when we reach either J z = 0 ( V c r i t = 1 for integer J), or J z = − 1 / 2 ( V c r i t = 1 / 2 for odd J). Accordingly, at such stage we have, independently of J [3]:

V c r i t = 1 / 2       for     half-integer     J       or         V c r i t = 1       for     integer     J . (18)

Let us insist: double inhabiting a p-site is not allowed. As a consequence, the Hamiltonian matrix’ size becomes ( 2 J + 1 ) × ( 2 J + 1 ) . Remark that the only manner to get different J values is to have such double occupancy [4]. Following [4], the J = N / 2 multiplet is the only one is to consider.

The free energy F and the partition function Z, with β the inverse temperature, and k B = 1 (Boltzmann) are expressed in the fashion

F = − T ln Z = − T ln T r a c e ( exp ( − β H ^ ) ) . (19)

Of course, in the Trace we sum over the J z quantum number m. Since H commutes with both J and J z we have a partition function Z

Z = ∑ m = − J m = J exp ( − β E m ) , (20)

where the energies E m are

E m = m − V ( J ( J + 1 ) − m 2 − J ) . (21)

The associated probabilities P m read [4]

P m = e x p ( − β E m ) Z , (22)

for all m = − J , − J + 1 , ⋯ , J − 1 , J . The entropy S is written as

S = − ∑ m = − J m = J   P m ln P m . (23)

The authors of Ref. [7] concocted a rather adequate functional F [ P ] of the probability distribution able to apprehend correlations in the manner of Shannon’s entropy grasping of randomness. This fact constituted a breakthrough, conceptualized with the help of the definition of López-Ruiz, Mancini, and Calbet (LMC) [7] of what became called since then the statistical complexity C, that individualized and quantified the relative contributions of entropy and structure to the new quantifier. The structural ingredient was described by a quantity they denominated disequilibrium D. This statistical complexity C has many applications today (for a small sample we mention Refs. [6] - [25]) C is zero in the two scenarios of perfect order and maximum disorder. LMC defined C as the product of Shannon’s entropy S and the disequilibrium D. D is a measure in probability space of the distance between the PD at hand and the uniform one. One has

C = S D . (24)

For our model, according to (23)), the uniform probabilities are P ( u ) = 1 / ( 2 J + 1 ) for all m between −J and J, so that for us here

D = ∑ m = − J m = J   ( P m − P ( u ) ) 2 , (25)

while

S = − ∑ m = − J m = J       P m ln P m . (26)

For more details regarding C we recommend Refs. [7]. With regards to this model, we always deal with a number of energy levels N = 2 J + 1 .

We deal with the results for the present two master equations (ME) above, that are to be compared to the quasi-static (st) results that arise out of considering, always, Boltzmann-Gibbs equilibrium-situations at the temperature T ( t ) for all t. As distance quantifiers we will employ two of them

· the Kullback-Leibler divergence D K L between two distinct probability densities (pertaining to the ME and the st treatments, respectively, and

· the entropy S.

If we generically call Q to any of these two quantifiers, the distances ME-st we are talking about here are of the form

d Q ( t ) = [ Q M E ( t ) − Q s t ( t ) ] 2 Q M E ( t ) 2 , (27)

D Q = ∫ 0 20 τ     d t   d Q ( t ) , (28)

where, obviously, the sub-index ME (or just “M”) refers to master equations results, and the sub-index “st” (or just “s”) to quasi-static ones.

We expect the distances from equilibrium to be sensitive to

1) Changes in the behavior of T with t and

2) Structural system’s changes with the coupling constant V,

3) The speed of the heating-up and cooling-off process, regulated by the parameter τ . The shorter τ , the faster the speed.

In Figure 1 we compare the time-evolution of the occupation probability P 0 (whose values are indicated in the y-axis), of the lowest lying level ‘‘0’’ in our three theoretical treatments, that we call, in abbreviate fashion, as Stat, ME, and GME. [Our probabilities are P 0 , P 1 , ⋯ , P 4 ] The coupling constant is V = 2 . Artifacts of the temperature-time triangle of apex T M are clearly noticeable for 0 < t < 2 τ , the triangle time-basis. For greater times rapid equilibration takes place, but in noticeably quicker fashion for GM than for M. Note the peak of P 0 in the ME treatment at t = 2 τ , where the temperature ceases changing.

Figure 2 is identical to Figure 1 save for the fact that here V = 0 (interaction-less system). The appearances are different, but still rapid equilibration takes place, but in quicker fashion for GM than for M. Of course, in an interaction-less system, added heat makes P 0 to diminish, while the fermion-fermion interaction in Figure 1 makes it to grow. The couple of Figure 1, Figure 2 beautifully illustrate the competition heat-interaction.