1. Introduction

The Boltzmann factor for a state with energy E and equilibrium inverse temperature β = 1/(k _B T) is given by

where T and k _B denote the equilibrium temperature and Boltzmann constant, respectively. When we regard the Boltzmann factor as a function in β, we know that it is a solution of the first order differential equation

where y = B _E (β).

In the thermal field theory [¹-³], the inverse temperature is regarded as an inverse time through the relation

Thus, the Boltzmann factor in the thermal field theory corresponds to the time evolution operator in the quantum field theory,

where the time evolution operator obeys the first order differential equation

Using Eq. (3), we know that the Eq. (5) is equivalent to Eq. (2).

The imaginary-time formalism is just a mathematical tool to represent the partition function as a path integral, which does not mean that the imaginary time is equivalent to temperature.

Use of the discrete time in physics has long history. First use was focussed on the discrete calculus of variations [⁴,⁵]. After this, the discrete Nöther theorem was found [⁶]. The possibility that time could be regarded as a discrete dynamical variable was examined through all phases of mechanics [⁷]. Some progress has been accomplished in this direction [⁴-¹⁰]. Recently, discrete time is adopted in the study of the stochastic wave equation [¹¹-¹⁶].

In the discrete time mechanics, the continuous time t is replaced with a discrete times forming the lattice

The classical discrete time mechanics was discussed in [¹⁰]. Here the derivative with respect to time in the continuous time mechanics was replaced with the finite difference between two adjacent discrete times, which is defined as

Thus, the quantum evolution operator in the discrete time lattice obeys the first order finite difference equation,

which is solved as

The Eq. (8) becomes Eq. (5) in the continuous time limit τ → 0 with t _n = nτ = t fixed.

In this paper we introduce the discrete inverse temperature from the discrete analogue of Eq. (4) and use it to construct the analogue of Boltzmann factor based on the discrete inverse temperature lattice. We investigate the discrete thermodynamics based on the discrete analogue of Boltzmann factor. We also discuss the superstatistics for the discrete inverse temperature. This paper is organized as follows: In Sec. 2 we discuss the discrete analogue of Boltzmann factor. In Sec. 3 we discuss the entropy and internal energy in discrete thermodynamics. In Sec. 4 we discuss some examples. In Sec. 5 we discuss the discrete superstatistics.

2. Discrete analogue of Boltzmann factor

From the discrete analogue of Eq. (3), we can define the discrete inverse temperature corresponding to the discrete time as

which defines the discrete inverse temperature lattice

Here we introduce the discrete inverse temperature as a discretization of the continuous inverse temperature. Among possible discrete inverse temperatures, we adopt the discrete inverse temperature with equal spacing for simplicity.

Another motivation of the discrete inverse temperature is to find the origin of Tsallis’s q-deformed Boltzmann factor. Thus, the discrete analogue of the inverse temperature is a kind of deformation corresponding to the Tsallis’s q-deformed Boltzmann factor, which will be discussed later.

For the discrete inverse temperatures, the discrete temperatures T _n are defined as

which implies

Thus, the above equation shows that (i) the discrete temperature forms a harmonic sequence, and (ii) it depends on the choice of β ₀. In this stage, three cases can be discussed:

Case of β ₀ > 0: In this case, we have non-zero minimum temperature. We have two values:

Case of β ₀ = 0: In this case, we have zero minimum temperature. We have two values:

Case of β ₀ < 0: In this case, we have negative temperature [¹⁷] for n<-β0/b. We have two values:

From now on we refer to the thermodynamics with a discrete Boltzmann factor as the discrete thermodynamics. In the discrete thermodynamics, the thermodynamic equilibrium occurs at the discrete inverse temperature. In this model, the thermodynamic quantities depends on the discrete inverse temperature. Thus, the discrete analogue of Eq. (2) is given by

where the finite difference is defined as

Solving Eq. (20), we get

which gives

where we set

The continuous version is obtained by setting β _n = β ₀ + nb = β and taking the limit n → ∞, b → 0 with nb + β ₀ = β fixed. In the continuous limit, we have the Boltzmann factor (1). Indeed, we get

When b is not zero, the Eq. (23) is well defined only for E ≤ 1/b. Thus, we demand that the energy is bounded from above for non-zero b. Thus, the discrete Boltzmann factor takes the form

This equation shows that the discrete Boltzmann factor is a kind of deformed Boltzmann factor. That means the existence of a relationship between the discrete Boltzmann factor and the well-known q-deformed Boltzmann factor appearing in Tsallis’s entropy [¹⁸].

Now, if we set β _N = β = finite, we have b = β/N. The continuous inverse temperature is then obtained by taking the limit b → 0 or N → ∞ with Nb = β fixed.

If we identify q - 1 = -b, we can express Eq. (26) in terms of q-exponential function [¹⁸] as

where q-exponential is defined as

and [ ]₊ denotes the Tsallis cutoff which implies that [x]₊ = 0 for x ≤ 0. Thus, the energy upper bound comes from the Tsallis cutoff. We note here that the Boltzmann factor given in Eq.(27) was introduced in Ref. [¹⁹].

In addition, we can see that Eq. (27) is not the same as the Tsallis’s q-deformed Boltzmann factor. But, when we identify N = 1/1 - q, we obtain

which is the Tsallis’s q-deformed Boltzmann factor.

From the discrete Boltzmann factor, we know that the probability finding a state with energy E at the equilibrium inverse temperature β _n for a system is proportional to the discrete Boltzmann factor,

where the partition function Z(βn) is defined as

for a discrete energy, and

for a continuous energy.

3. Entropy and internal energy in discrete thermodynamics

For a system which is in thermal equilibrium with a reservoir of given discrete inverse temperature β _n we use the canonical ensemble formulation. The probability of finding a state with E _i in this system is given by

which obeys

Now let us consider the entropy in the form,

For the average, the linear average is not the only one that can be used. In the general theory of averages, for any monotonously increasing and bijective function ψ(x) called deforming map, the Kolmogorov-Nagumo (KN) average [²⁰, ²¹] can be adopted. Now let us define the internal energy U as the KN mean of energy as

Based on the variational method for the discrete theory [⁴,⁵], we can obtain the maximum entropy probability form the extreme condition of the functional,

Demanding ∂Φ/∂Pi=0, we get

Comparing Eq. (32) with Eq. (37), we get

and

Thus, we have

The internal energy (or energy expectation value) is then given by

Using the relation

we have

In the continuum limit, Eq. (44) becomes

The discrete specific heat for the temperature interval Jn=Tn+1,Tn is defined as

Thus, the Boltzmann-Gibbs relation reads

4. Some examples

Now let us discuss some examples based on the discrete thermodynamics.

5. Discrete superstatistics

Now let us discuss the superstatistics for the discrete Boltzamnn factor. In this case we regard as the discrete inverse temperatures as random variables β~m (m = 0,1,2,…) which belongs to 𝔹. Besides, we demand that the mean of β~m is the equilibrium inverse temperature β _n and set the variance of β~m to be σ ². For the random inverse temperatures, we introduce the probability distribution function f(β~m) obeying

and

and

From the superstatistics [²²], we define the superstatistical Boltzmann factor as

The superstatistical partition function is defined as

As an example, let us consider the following probability distribution function

From Eq. (69), Eq. (70) and Eq. (71), we get

Thus, we have

From the superstatistics [²²], we define the superstatistical discrete Boltzmann factor as

In the low-energy asymptotics of superstatistics, the generalized Boltzmann factor is then given by

and the superstatistical partition function is given by

6. Conclusion

Based on the discrete time mechanics and inverse temperature formalism in the thermal field theory, we introduced the discrete inverse temperature and discussed the discrete analogue of the thermodynamics. To do so, we introduced the discrete inverse temperature lattice and defined the discrete analogue of Boltzmann factor. For a general discussion, we considered that there exists a finite minimum inverse temperature corresponding to a finite maximum temperature. We adopted the discrete inverse temperature as having the equal spacing.

Our discrete inverse temperature is different from the discrete inverse temperature [²³] appearing in the superstatistics where discrete inverse temperature was regarded as a fluctuation around the equilibrium temperature. But our discrete inverse temperature can be regarded as a discretization of the continuous inverse temperature. The discrete inverse temperature in Eq. (11) is not a unique choice. We can consider more general discrete inverse temperature. For example, we can consider the pseudo discrete inverse temperature through

or

where the pseudo-addition and pseudo subtraction [²⁴] are defined as

and

For simplicity, in this paper we considered the case of f = Id.

We found that discrete Boltzmann factor could be obtained for the maximum entropy principle with a certain KN mean for the energy. We found the formula for the internal energy defined as a KN mean of energy and investigated the relation between the internal energy and partition function. As examples we presented three examples; classical harmonic oscillator, Maxwell-Boltzmann distribution of molecular speeds in a gas, two level system. Finally we discussed the discrete superstatistics for the discrete thermodynamics.