Temperature distribution in finite systems: application to the one-dimensional Ising chain

Abstract

Thermodynamic studies of small systems interacting with a finite environment display an interesting statistical behavior, similar to complex non-equilibrium systems. In both situations there are several applicable definitions of inverse temperature, either intrinsic or dependent of the statistical ensemble, and uncertainty in these quantities has to be taken into account. In this work we develop these concepts using as an example an isolated one-dimensional Ising chain subsystem that does not follow the canonical distribution. In the context of this example, we explicitly show that the theory of superstatistics cannot describe the behavior of the subsystem, and verify a recently reported relation between the ensemble and microcanonical inverse temperatures. Our results hint at a new framework for dealing with regions of microcanonical systems with positive heat capacity, which should be described by some new class of statistical ensembles outside superstatistics but still preserving the notion of temperature uncertainty.

Appendix A: Derivation of the ensemble temperature of the Ising chain

In order to show the step-by-step computation of how the ensemble inverse temperature $\beta _F$ was obtained, consider the following definitions which relate the original parameters with the energy of the system E and the sub-system $E_s$, that is,

$$\begin{aligned} \frac{M}{N} = \frac{1}{2}+\frac{E}{2N}, \end{aligned}$$

(A1)

$$\begin{aligned} K = \frac{1}{2}\left( \gamma N - 1 + E_s\right) , \end{aligned}$$

(A2)

and $L=\gamma N$. We start with the quantity $\omega _n$ of Ref. [24], which in our case corresponds to the ensemble function as in Eq. 27.

The next step was to take the logarithmic derivative of $\rho$ in Eq. 27, with respect to the sub-system energy,

$$\begin{aligned} \frac{\partial \ln \rho (E_s)}{\partial E_s}&= \frac{1}{2}\Bigg [\psi \left( \frac{1}{2}\big (3+E-E_s+ N(1-\gamma )\big )\right) \nonumber \\&\quad - \psi \left( \frac{1}{2}\big (3-E+E_s+ N(1-\gamma )\big )\right) \Bigg ] \end{aligned}$$

(A3)

Applying the definition of $\beta _F$ and simplifying, we obtained the last expression in terms of the harmonic number $H_n$, finally we obtained a simplified expression for $\beta _{F}$

$$\begin{aligned} \beta _{F} =&\frac{1}{2}\ln {\Bigg [\frac{N(1-\gamma )+(E_s-E)}{N(1-\gamma )+(E-E_s)}\Bigg ]}, \end{aligned}$$

(A4)

that, when cancelling N and using $\epsilon _s \mathrel {\mathop :}=E_s/N$, $\epsilon \mathrel {\mathop :}=E/N$ reduces to Eq. 31.