Holographic equipartition and the maximization of entropy

The accelerated expansion of the Universe can be interpreted as a tendency to satisfy holographic equipartition. It can be expressed by a simple law, $\mathrm{\Delta }V=\mathrm{\Delta }t(N_{\mathrm{surf}}-\epsilon N_{\text{bulk}})$, where $V$ is the Hubble volume in Planck units, $t$ is the cosmic time in Planck units, and $N_{\mathrm{surf}/\text{bulk}}$ is the number of degrees of freedom on the horizon/bulk of the Universe. We show that this holographic equipartition law effectively implies the maximization of entropy. In the cosmological context, a system that obeys the holographic equipartition law behaves as an ordinary macroscopic system that proceeds to an equilibrium state of maximum entropy. We consider the standard $\mathrm{\Lambda }\mathrm{CDM}$ model of the Universe and show that it is consistent with the holographic equipartition law. Analyzing the entropy evolution, we find that it also proceeds to an equilibrium state of maximum entropy.

Figure 1

Evolution of the holographic discrepancy with the progress in cosmic time.

Figure 2

Evolution of the entropy of the horizon. The vertical axis shows the horizon entropy in units of $k_B$.

Figure 3

Variation of $S^{\prime }$ against $H_{0}t$. The vertical axis shows the rate of change of entropy with the progress of cosmic time in units of $k_B$.

Figure 4

Variation of $S^{\prime \prime }$ against $H_{0}t$. The vertical axis shows the variation of the second derivative of the entropy with the progress of cosmic time in units of $k_B$.