Consistent thermodynamic framework for interacting particles by neglecting thermal noise

An effective temperature $\theta$, conjugated to a generalized entropy $s_q$, was introduced recently for a system of interacting particles. Since $\theta$ presents values much higher than those of typical room temperatures $T\ll \theta$, the thermal noise can be neglected ($T/\theta \simeq 0$) in these systems. Moreover, the consistency of this definition, as well as of a form analogous to the first law of thermodynamics, $du=\theta d{s}_q+\delta W$, were verified lately by means of a Carnot cycle, whose efficiency was shown to present the usual form, $\eta =1-({\theta }_2/{\theta }_1)$. Herein we explore further the heat contribution $\delta Q=\theta d{s}_q$ by proposing a way for a heat exchange between two such systems, as well as its associated thermal equilibrium. As a consequence, the zeroth principle is also established. Moreover, we consolidate the first-law proposal by following the usual procedure for obtaining different potentials, i.e., applying Legendre transformations for distinct pairs of independent variables. From these potentials we derive the equation of state, Maxwell relations, and define response functions. All results presented are shown to be consistent with those of standard thermodynamics for $T>0$.

Figure 1

A thermal contact may be achieved by considering a movable, rigid, and impermeable wall along the $x$ direction (dashed line) separating two systems 1 and 2, characterized by $N_1$ and $N_2$ vortices, respectively. In case (a) a typical situation is represented where the net force exerted by the vortices of system 2 on the wall is greater than the one of the vortices in system 1, so the wall is pushed upwards. A mechanical equilibrium is reached in (b), where the forces from both sides cancel. The transformation is given by a change in the lengths in the $y$ direction, from an initial state defined by lengths $(L_{y,i}^{\left(1\right)},L_{y,i}^{\left(2\right)})$, to a final state defined by lengths $(L_{y,f}^{\left(1\right)},L_{y,f}^{\left(2\right)})$.

Figure 2

The Carnot refrigerator for a system of interacting vortices under overdamped motion by neglecting thermal noise. The abscissa $\alpha {\lambda }^2$ presents dimensions of energy (e.g., Joules), whereas the ordinate $\sigma /{\lambda }^2$ is dimensionless. The transformations for $\sigma$ constant are adiabatic, and herein they were chosen to occur for $(\sigma /{\lambda }^2)=0.45$ ($a\to b$) and $(\sigma /{\lambda }^2)=0.25$ ($c\to d$). The isothermal transformations are characterized by $\sigma \sim {\alpha }^{-2/3}$ [cf. Eq. (2.19)] and they occur for $k{\theta }_1=5$ (units of energy) in $b\to c$, and $k{\theta }_2=1$ (units of energy) in $d\to a$, i.e., ${\theta }_1>{\theta }_2$. Operating as a refrigerator, an amount of heat $Q_2$ is absorbed at ${\theta }_2$ and $Q_1$ is released at ${\theta }_1$; as shown in text, the area inside the cycle represents the total work $W$ done $on$ the refrigerator with $Q_1=W+Q_2$ (conventionalizing all these three quantities as positive). The cycle above holds for any system of units, e.g., by considering all quantities with dimensions of energy in Joules.

Figure 3

The Otto cycle for a system of interacting vortices under overdamped motion by neglecting thermal noise. The abscissa $\alpha {\lambda }^2$ presents dimensions of energy (e.g., Joules), whereas the ordinate $\sigma /{\lambda }^2$ is dimensionless. The transformations for $\sigma$ constant are adiabatic, and herein they were chosen to occur for $(\sigma /{\lambda }^2)=0.45$ ($b\to c$) and $(\sigma /{\lambda }^2)=0.25$ ($d\to a$). The iso-$\alpha$ transformations were chosen to occur for $\alpha {\lambda }^2=3.0$ (units of energy) ($a\to b$) and $\alpha {\lambda }^2=1.0$ (units of energy) ($c\to d$), respectively. 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. (2.17). The abscissa $\alpha {\lambda }^2$ presents dimensions of energy, whereas the ordinate $\sigma /{\lambda }^2$ is dimensionless; the cycle above holds for any system of units, e.g., one may consider all quantities with dimensions of energy in Joules.