Repulsive particles under a general external potential: Thermodynamics by neglecting thermal noise

A recent proposal of an effective temperature $\theta$, conjugated to a generalized entropy $s_q$, typical of nonextensive statistical mechanics, has led to a consistent thermodynamic framework in the case $q=2$. The proposal was explored for repulsively interacting vortices, currently used for modeling type-II superconductors. In these systems, the variable $\theta$ presents values much higher than those of typical room temperatures $T$, so that the thermal noise can be neglected ($T/\theta \simeq 0$). The whole procedure was developed for an equilibrium state obtained after a sufficiently long-time evolution, associated with a nonlinear Fokker-Planck equation and approached due to a confining external harmonic potential, $\varphi \left(x\right)=\alpha x^2/2$ ($\alpha >0$). Herein, the thermodynamic framework is extended to a quite general confining potential, namely $\varphi \left(x\right)={\alpha \left|x\right|}^z/z$ ($z>1$). It is shown that the main results of the previous analyses hold for any $z>1$: (i) The definition of the effective temperature $\theta$ conjugated to the entropy $s_2$. (ii) The construction of a Carnot cycle, whose efficiency is shown to be $\eta =1-({\theta }_2/{\theta }_1)$, where ${\theta }_1$ and ${\theta }_2$ are the effective temperatures associated with two isothermal transformations, with ${\theta }_1>{\theta }_2$. The special character of the Carnot cycle is indicated by analyzing another cycle that presents an efficiency depending on $z$. (iii) Applying Legendre transformations for a distinct pair of variables, different thermodynamic potentials are obtained, and furthermore, Maxwell relations and response functions are derived. The present approach shows a consistent thermodynamic framework, suggesting that these results should hold for a general confining potential $\varphi \left(x\right)$, increasing the possibility of experimental verifications.

Figure 1

(a) Isothermal transformations are exhibited in the plane $\sigma /{\lambda }^z$ (dimensionless) vs $\alpha {\lambda }^z$ (dimensions of energy) for typical values of $z$; accordingly, the dimensions of both quantities $\sigma$ and $\alpha$ vary with $z$ in such a way that the product $\sigma \alpha$ presents dimensions of energy. (b) The Carnot cycle $a\to b\to c\to d\to a$ is represented in the particular case $z=4$. The transformations for $\sigma$ constant are adiabatic, and herein they were chosen to occur for $(\sigma /{\lambda }^4)=0.45$ ($b\to c$) and $(\sigma /{\lambda }^4)=0.25$ ($d\to a$). The isothermal transformations are characterized by $\sigma \sim {\alpha }^{-4/5}$ [cf. Eq. (3.3)], and they occur for $k{\theta }_1=5$ (units of energy) in $a\to b$, and $k{\theta }_2=1$ (units of energy) in $c\to d$, i.e., ${\theta }_1>{\theta }_2$. The area inside the cycle represents the total work $W$ done on the system, which is negative, as expected from Eq. (3.1). The cycle above holds for any system of units, e.g., one may consider all quantities with dimensions of energy in Joules.

Figure 2

The Otto cycle is represented for the particular case $z=4$; the cycle is composed of two adiabatic ($\sigma$ constant) transformations and two iso-$\alpha$ ($\alpha$ constant) transformations, intercalated. The abscissa $\alpha {\lambda }^4$ presents dimensions of energy (e.g., Joules), whereas the ordinate $\sigma /{\lambda }^4$ is dimensionless. An amount of heat $Q_1$ is absorbed at transformation $a\to b$, whereas $Q_2$ is released at $c\to d$. The area inside the cycle represents the total work $W$ done on the system, which is negative, as expected from Eq. (3.1). The cycle above holds for any system of units, e.g., one may consider all quantities with dimensions of energy in Joules.