The model advanced in Ref. [3] and further discussed in Ref. [4], considers M fermions distributed amongst (2M)-fold degenerate single-particle (sp) levels, separated by a sp energy gap ϵ . Two quantum numbers μ and p are attached to a generic single particle state. The first adopts the values μ = − 1 (lower level) and μ = + 1 (upper level). p, usually called quasi-spin or pseudo spin, singles out a state within the M-fold degeneracy. The couple p , μ can also be viewed as a ‘‘site’’ that is either occupied or empty. One has

M = 2 J , (3)

where J denotes an “angular momentum’’. One introduces now the quasi-spin operators

J ^ + = ∑ p     C p , + † C p , − , (4)

J ^ − = ∑ p     C p , − † C p , + , (5)

J ^ z = ∑ p , μ     μ C p , μ † C p , μ , (6)

J ^ 2 = J ^ z 2 + 1 2 ( J ^ + J ^ − + J ^ − J ^ + ) , (7)

the eigenvalues of J ^ 2 being of the form J ( J + 1 ) . The Hamiltonian of reference [3], a spin-flip one, reads

H ^ = ϵ J ^ z − V s ( 1 2 ( J ^ + J ^ − + J ^ − J ^ + ) − J ^ ) , (8)

or, with V = V s / ϵ (equivalently, take ϵ = 1 ),

H ^ = J ^ z − V ( 1 2 ( J ^ + J ^ − + J ^ − J ^ + ) − J ^ ) , (9)

so that the unperturbed ground state (ugs) ( V = 0 ) becomes, according to Equation (3),

| J , J z 〉 = | J , − M / 2 〉 , (10)

whose energy is

E o = − M / 2 . (11)

Doubly occupied p-sites are not allowed for. The Hamiltonian commutes with J ^ 2 and J ^ z . This entails that the exact solution will belong to the J-multiplet of the unperturbed ground state. The multiplet’s states are denoted as | J , m 〉 , and one of them will minimize the energy. The associated m value of this state will depend upon the coupling strength V of the interaction. As we just mentioned, for the ugs one has m = − J = − M / 2 . Evidently, the interaction-operator ( J ^ + J ^ − + J ^ − J ^ + ) is a quasi-spin flipping operator. Thus, this operator becomes the more ‘‘effective’’ the more balanced the populations of the two-levels become.

As V grows from zero, the ugs energy E o is not immediately affected. It conserves its value till a critical V-specific value is attained, of 1 / ( M − 1 ) . At this juncture, the interacting gs suddenly turns out to be | J , − M / 2 + 1 〉 . If V continues its growth, new phase transitions (PT) take place. The PT between J z = − k and J z = − k + 1 ensues at V = 1 / ( 2 k − 1 ) . The successive PT’s processes ends up when we attain 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). Thus, at such juncture one has, 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 . (12)

To repeat: double occupancy of a p-site is not permitted. Thus, the Hamiltonian matrix’ size is ( 2 J + 1 ) × ( 2 J + 1 ) . The only way to get different J’s is to have double occupancy [4]. Following this last reference, the J = N / 2 multiplet is the only one we need to consider.

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

F = − T l n Z = − T l n T r a c e ( e x p ( − β H ^ ) ) . (13)

In the Trace we sum over the J z quantum number m. As H commutes with both J and J z one finds

Z = ∑ m = − J m = J exp ( − β E m ) , (14)

where the energy E m are

E m = m − V ( J ( J + 1 ) − m 2 − J ) . (15)

Our all important pertinent probabilities P m are [4]

P m = e x p ( − β E m ) Z , (16)

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

S = − ∑ m = − J m = J     P m ln P m . (17)

Sometimes one wishes to grab hold of a system’s correlation structures just as entropy grasps disorder. Why? Because such correlations strongly influence the main features of the prevailing PD describing physical processes. It is obvious that the opposite extremes of perfect order and maximal randomness do not manifest notable structural correlations [6]. In between these two instances, a wide range of structural degrees (SD) usually exist, that are in turn reflected by the traits of the prevailing PD P.

The authors of Ref. [6], invented a quite adequate functional F [ P ] that can apprehend correlations just as Shannon’s entropy encapsulates randomness, which indeed constituted an important breakthrough. Their ideas were conceptualized via the definition of López-Ruiz, Mancini, and Calbet (LMC) [6] of what became called the statistical complexity C.

LMC C individualized and quantified the respective contributions of entropy and structure. The last one was described by a quantity called disequilibrium D. Their concept of statistical complexity C was widely accepted (for a sample see, for instance, Refs. [6] - [27] ). C vanishes in the two situations of perfect order and maximum disorder, being defined as the product of Shannon’s entropy S and the disequilibrium D. More specifically, the latter is a measure in probability space of the distance measure the prevailing PD and the uniform PD, so that one writes

C = S D , (18)

with (see (17)), in our case, the uniform probabilities are P ( u ) = 1 / ( 2 J + 1 ) for all m between − J and J, so that, the LMC disequilibrium is

D = ∑ m = − J m = J ( P m − P ( u ) ) 2 , (19)

while

S = − ∑ m = − J m = J   P m ln P m . (20)

For details, properties, and others applications of C, see Refs. [6].

We consider that our system is at the equilibrium temperature T 0 at t = 0 . Then he system is heated and reaches a temperature T M at t = τ . At this stage, the systems cools-off, reaching a temperature T f = T 0 at time t = 2 τ . This is the temperature of the heat-reservoir at all later times.

Inspired by the 2-level systems treatment of [2], we tackle now a system of N levels l i ( i = 0 , 1 , 2 , ⋯ ) , with level-energies E 0 , E ! , etc., and selections rules that allow for only certain kinds of transitions. In the usual parlance of quantum many body theories, we permit only transitions from a state with k-particle-holes (p-h) to ones with either k + 1 or k − 1 p-h, so that

d p n d t = p n − 1 exp ( − β E n ) − p n exp ( − β E n − 1 ) , (21)

and we follow in this way till we face, for the last 3 steps,

d p 2 d t = p 1 exp ( − β E 2 ) − p 2 exp ( − β E 1 ) ,

d p 1 d t = p 0 exp ( − β E 1 ) − p 1 exp ( − β E 0 ) ,

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 ) ] .

The last equation guarantees normalization of probabilities.

The initial conditions are p i ( 0 ) = e x p ( − E i / k T ) / Z for all i.

β depends upon time, as described at the beginning of this Section and we work with

β ( t ) = 1 / T ( t ) ,

and

E j = j ;       all     j .

If β were constant, then the time-dependent probabilities p i ( t ) would relax to stationary Gibbs-distributions.

The numerical calculations consider the cases of N = 3,4,5 , 6 , corresponding respectively to J = 1 , 3 / 2 , 2 , 5 / 2 for our model above.

We deal with the results of our master equation (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

• The probabilities’ differences 1) Global D P = ∑ m [ P m M E ( t ) − P m s t ( t ) ] 2 / ( P m M E ( t ) ) 2 , and 2) Individual d P m = [ P m M E ( t ) − P m s t ( t ) ] 2 / ( P m M E ( t ) ) 2 ,

• the entropy S,

• the free energy F,

• the mean energy U, and

• the statistical complexity C.

If we generically call Q to any of these quantifiers, the distances are of the form

d Q ( t ) = [ Q M E ( t ) − Q s t ( t ) ] 2 Q M E ( t ) 2 , (22)

D Q = ∫ 0 20 τ   d t   d Q ( t ) , (23)

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

We expect the distances 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.

Note that, with reference to our model above, we have N = 2 J + 1 . Figure 2 refers to N = 3 and the individual probabilities-distance D P m vs. t. It is clearly seen that there is much sensitivity to the details of the heating process. Figure 3 refers also to N = 3 but with the global D P vs. V. Here the ensuing picture is of a more complex nature. There is a system’s phase transition (PT) at V = 1 , and our distance tends to decrease as we approach the PT. Thus we see that D P is sensitive to the internal dynamics of the system. This is no trivial issue. Why should the OE-equilibrium distance behave in such a manner? This is a new fact discovered here, as far as we know. But the picture becomes even more complex when we consider the speed of the heating process. If it is large enough, the distance OE-E grows again with V after the PT, but if it is low enough, it continues

decreasing as V grows. We conclude that the distance quantifiers is sensitive both to the system’s internal dynamics and to the details of the heating process.

Starting now with N = 4 we pass to watch in Figure 4 the performance with time of the other quantifiers, of thermal origin, namely S, F, U, and C. As compared with the probabilities-distance, the thermal-distance (TD) is much more sensitive to the details of the heating process. When it finishes, at t = 2 τ , the four different TD vanish. We pass now to Figure 5 so as to analyze, for N = 3 ,

the behavior of the four thermal distances. We see that each of them behaves in a distinctive way, that forces individual consideration.

• S-distance: it diminishes as V grows from zero till reaching the phase transition (PT). After wards it grows again as V continues increasing. At the PT the distance is minimal. The after-PT grows is the more pronounced the speedier the heating process (HP). Even after it has finished, S seems to keep memory of it, since its growth differentiates amongst the different τ ’s.

• F-distance: It senses the PT if the HT is fast. Otherwise, it does not. It clearly is able fo distinguish amongst the different τ ’s.

• U-distance: tends to vanish as V grows. It is rather indifferent both to the τ -value and to the PT.

• C-distance: this is supposed to be the most sophisticated statistical of our four tools. Here it only tells us that the difference between the off-equilibrium complexity and the equilibrium one tend to become identical as the coupling constant grows. In other words, if the system is tightly bound, whether one heats it up or not becomes less and less relevant as the bonding augments, which sounds reasonable.

Figure 6 is identical to Figure 5, except for the fact that N is larger and equals 5. There are two, not just one, PTs now, both for V < 1 , that do not seem to leave any trace in the graphs. The differences between what the different thermal indicators say is smaller here. All of them indicate that if the system is tightly bound, whether one heats it up or not becomes less and less relevant as the bonding augments, as we saw above for N = 3 , but only in the C-case. Here, once again, the statistical complexity appears to be the mpst sensitive of the indicators, as the C-distance is the only one of the four here used able to detect differences amongst the distinct τ values. These conclusions are reinforced if

we confront now the case N = 6 . Emphasize that C is undoubtedly the most sensitive quantifier (Figure 7).

• The first, rather surprising observation, is that our systems lose structural details as N grows.

• The time-evolution of the distance between the two pertinent probability distributions is quite sensitive to the heating-cooling process.

• The shorter the heating-coolong period, the more sensitive the probabilities-distance quantifier becomes to the internal systems’ structure, as revealed by the ground-state phase transition.

• The four thermal distances time-evolutions are also quite sensitive to the heating-cooling process.

• The four thermal distances evolutions with a growing coupling constant are quite sensitive to the internal dynamics and to g the heating-cooling details for N = 3 .

• This sensitivity is gradually lost as N grows.

• For N ≥ 4 , the four thermal distances evolutions with a growing coupling constant rapidly vanish. The strongly interacting systems seem no to care whether it is heated or cooled.

• This entails that, the larger the coupling constant V, the more rapidly equilibrium is attained, as evidences by the diminution of the values of our thermal distance quantifiers. However, C is the most sensitive of the four indicators.

The authors declare no conflicts of interest regarding the publication of this paper.

Plastino, A.R., Ferri, G.L., Rocca, M.C. and Plastino, A. (2020) Information-Based Numerical Distances between Equilibrium and Non-Equilibrium States. Journal of Modern Physics, 11, 1031-1043. https://doi.org/10.4236/jmp.2020.117065