Thermodynamic entropy and chaos in a discrete hydrodynamical system

We show that the thermodynamic entropy density is proportional to the largest Lyapunov exponent (LLE) of a discrete hydrodynamical system, a deterministic two-dimensional lattice gas automaton. The definition of the LLE for cellular automata is based on the concept of Boolean derivatives and is formally equivalent to that of continuous dynamical systems. This relation is justified using a Markovian model. In an irreversible process with an initial density difference between both halves of the system, we find that Boltzmann’s $H$ function is linearly related to the expansion factor of the LLE although the latter is more sensitive to the presence of traveling waves.

Figure 1

All these states have the same number of particles, momentum, and energy. An arrow represents the presence of a particle with the velocity of the arrow, and the open circle represents a particle at rest.

Figure 2

In this example, the input state at some fixed site is the one shown on the left of Fig. 1 and in the column number assigned to this site, the output state is the fourth one of that figure. These are shown in the upper part of this figure. Then, by inverting the velocities we get the state on the lower left and the collision table should contain in the same column for this input state the output state shown on the lower right that upon inversion of the velocities yields the original incoming state.

Figure 3

Distribution functions in equilibrium as a function of $e$ for $n=4$. The continuous curves are the calculated values, and the symbols are the results of numerical simulations. For $e=e_F$ there is a rest particle in all sites and no fast particles moving along the diagonals. The other three particles can occupy one of the four slow directions. For $e=e_M$ there are no rest or slow particles, the four particles are moving along the diagonals.

Figure 4

Entropy density $s$ (solid line) and LLE $\lambda$ (dashed line) for $n=4$, simulations with $R=40$ in a $40\times 40$ lattice.

Figure 5

We show $s$ versus $\lambda$ for $n=2$ (diamonds), $n=4$ (solid line), $n=4.5$ (dotted line), and $n=6$ (pluses). The simulations are performed for different values of $e$ from $e_F\left(n\right)$ to $e_M\left(n\right)$.

Figure 6

Boltzmann’s function $H$ (thick line) and Lyapunov expansion factor $⟨\ell ⟩$ (oscillating thin line) versus time $t$, simulations with $n_L=7.2$, $e_L=4.8$, $n_R=1.8$, $e_R=1.2$, and $R=40$ on a $40\times 40$ lattice. The inset shows that, disregarding the oscillations of $⟨\ell ⟩$, there is a linear relation between these quantities. The dashed line is the best fit $H=1.15⟨\ell ⟩+0.2$.