Specific heat and entropy of $N$-body nonextensive systems

We have studied finite $N$-body $D$-dimensional nonextensive ideal gases and harmonic oscillators, by using the maximum-entropy methods with the $q$ and normal averages ($q$: the entropic index). The validity range, specific heat and Tsallis entropy obtained by the two average methods are compared. Validity ranges of the $q$- and normal averages are $0<q<q_U$ and $q>q_L$, respectively, where $q_U=1+{\left(\eta DN\right)}^{-1}$, $q_L=1-{\left(\eta DN+1\right)}^{-1}$ and $\eta =1/2$ $\left(\eta =1\right)$ for ideal gases (harmonic oscillators). The energy and specific heat in the $q$ and normal averages coincide with those in the Boltzmann-Gibbs statistics, although this coincidence does not hold for the fluctuation of energy. The Tsallis entropy for $N\left|q-1\right|⪢1$ obtained by the $q$ average is quite different from that derived by the normal average, despite a fairly good agreement of the two results for $\left|q-1\right|⪡1$. It has been pointed out that first-principles approaches previously proposed in the superstatistics yield additive $N$-body entropy $\left(S^{\left(N\right)}=N{S}^{\left(1\right)}\right)$ which is in contrast with the nonadditive Tsallis entropy.

Figure 1

(Color online) $S_q/k_{B}N$ of ideal gases as a function of $q$ for $N=100$ calculated by the $q$ average ($q$-av.: the dashed curve) and the normal average (N-av.: the solid curve), and the difference between the two entropies (the bold curve) ($D=1$, $T/T_0=1.0$: $T_0=h^2/k_{B}m$). The inset shows the validity range of the two averages in the $q\text{−}N$ space: the MEMs with the $q$- and normal averages are valid for $0<q<q_U$ and $q>q_L$, respectively, and for $q_L<q<q_U$ both the averages are valid: the dashed line denotes $N=100$.

Figure 2

(Color online) $S_q/k_{B}N$ as functions of $q$ and $N$ obtained by the $q$-average $\left(D=1,T/T_0=1.0\right)$: note the logarithmic scale in the ordinate and abscissa for $N$.

Figure 3

(Color online) $S_q/k_{B}N$ as functions of $q$ and $T$ obtained by the $q$-average $\left(D=1,N=100\right)$: note the logarithmic scale in the ordinate.

Figure 4

(Color online) $S_q/k_{B}N$ as functions of $T$ and $N$ obtained by the $q$-average $\left(D=1,q=0.8\right)$: note logarithmic scales in the ordinate and abscissa for $N$.

Figure 5

(Color online) ${\tilde{S}}_q/k_{B}N$ as functions of $q$- and $N$ obtained by the normal average $\left(D=1,T/T_0=1.0\right)$: note the logarithmic scale in the abscissa for $N$.

Figure 6

(Color online) ${\tilde{S}}_q/k_{B}N$ as functions of $q$ and $T$ obtained by the normal average $\left(D=1,N=100\right)$.

Figure 7

(Color online) ${\tilde{S}}_q/k_{B}N$ as functions of $T$ and $N$ obtained by the normal average $\left(D=1,q=1.2\right)$: note the logarithmic scale in the abscissa for $N$.

Figure 8

(Color online) The relative fluctuation of $\Delta U_q/U_q$ as functions of $q$ and $\left(\eta DN\right)$ calculated by the $q$ average: $\eta =1/2\left(\eta =1\right)$ for ideal gases (harmonic oscillators).