Boltzmann Entropy for Dense Fluids Not in Local Equilibrium

Using computer simulations, we investigate the time evolution of the (Boltzmann) entropy of a dense fluid not in local equilibrium. The macrovariables $M$ describing the system are the (empirical) particle density $f=\{f(\underline{x},\underline{v})\}$ and the total energy $E$. We find that $S(f_t,E)$ is a monotone increasing in time even when its kinetic part is decreasing. We argue that for isolated Hamiltonian systems monotonicity of $S(M_t)=S(M_{X_t})$ should hold generally for “typical” (the overwhelming majority of) initial microstates (phase points) $X_0$ belonging to the initial macrostate $M_0$, satisfying $M_{X_0}=M_0$. This is a consequence of Liouville’s theorem when $M_t$ evolves according to an autonomous deterministic law.

Figure 1

Evolution of $S^{(m)}/V$, $\Phi$, and $S/V$ following the initial nonequilibrium state. The particles interact with a cutoff Lennard-Jones potential and $N=90000$. The energy density and the initial potential energy density are $e=0.6$ and $\Phi =-0.5917\dots$.

Figure 2

Evolution of $S^{(m)}/V$, $k$, $\Phi$, and $S/V$ for different size systems $N=100,400,1600,90000$, and ${10}^6$. The particles interact with a cutoff $r^{-6}$ potential. The energy density and the initial potential energy density are $e=0.7$ and $\Phi =0.2049$.

Figure 3

Evolution of $f_t(v)$ for three different initial microscopic configurations. With $f_0$ and $E$ the particles interact with a truncated $r^{-6}$ potential and $N=90000$.

Figure 4

Evolution of $f_t(v)$ for an initial condition with all particles in a square lattice and equal speeds with random directions for $n=0.5$, $e=0.7$, $\overline{\varphi }(r)=r^{-6}$, and $N=90000$. Full dots and empty circles are the values of $f_t$ corresponding to two different microscopic states, respectively.

Figure 5

Evolution of the nonequilibrium entropy $S_{(2)}[f,E]/L$ for a one-dimensional system with $N={10}^5$ particles with alternating masses $m_1=1$ and $m_2=(1+\sqrt{5})/2$. Initially the particles with mass $m_1$ ($m_2$) have a Maxwellian velocity distribution with temperature $T_1=1$ ($T_2=5$). $S_{\mathrm{g}\mathrm{a}\mathrm{s}}(f_i)$ is the partial entropy for the $i$th species.