Thermodynamics aspects of noise-induced phase synchronization

In this article, we present an approach for the thermodynamics of phase oscillators induced by an internal multiplicative noise. We analytically derive the free energy, entropy, internal energy, and specific heat. In this framework, the formulation of the first law of thermodynamics requires the definition of a synchronization field acting on the phase oscillators. By introducing the synchronization field, we have consistently obtained the susceptibility and analyzed its behavior. This allows us to characterize distinct phases in the system, which we have denoted as synchronized and parasynchronized phases, in analogy with magnetism. The system also shows a rich complex behavior, exhibiting ideal gas characteristics for low temperatures and susceptibility anomalies that are similar to those present in complex fluids such as water.

Figure 1

Spectrum of order parameter $r$ in space $(T_{\mathrm{eff}},\sigma )$. The black region above $T_{\mathrm{eff}}=T_{\mathrm{eff}c}=1$ defines the region without order $r=0$. Note that the region $T_{\mathrm{eff}c}<1$ contains the entire synchronized region including that for which $T>1$; we call this as a parasynchronized phase.

Figure 2

Order parameter $r$ as a function of noise coupling $\sigma$ for isotherms $T=1, T>1$, and $T<1$. The region $T>1$ characterizes the parasynchronized phase with $\sigma <0$, and the region $T\le 1$ corresponds to the synchronized phase.

Figure 3

Order parameter $r$ as a function of temperature $T$ for values of $\sigma$. Note that all curves display a typical mean-field behavior.

Figure 4

Entropy as a function of temperature $T$ for values of $\sigma$. Note that all entropy reaches the maximum value at $ln\left(2\pi \right)$, which is the full incoherent state. All entropy becomes $S=-\infty$ at $T=0$, which is the full state synchronizing.

Figure 5

Specific heat $C_{\sigma }\left(T\right)$ as a function of temperature $T$ for values of $\sigma$. All curves start at $C_{\sigma }=\frac{1}{2}$ for $T=0$ and grow until they reach the maximum value at $T_{\mathrm{eff}c}=1$. For $T>T_{\mathrm{eff}c}, C_{\sigma }=0$, as expected.

Figure 6

Order parameter $r$ as a function of external field $H_{\sigma }$ for several isotherms $T$. All curves saturate at $r=1$, analogous to a magnetic system. The curve $T=1$ separates the synchronized phase $T\le 1$ from the parasynchronized phase $T>1$.

Figure 7

Inverse of susceptibility ${\chi }^{-1}$ as a function of external field $H_{\sigma }$ for fixed isotherms $T$. Curves $T=1.3$ and $T=1$ correspond to the parasynchronized phase. Curve $T=0.7$ corresponds to the synchronized phase. Note that for critical temperature $T=1$ and null field $H_{\sigma }\to 0, \chi ={(T_c-T)}^{-1}$ diverges, as expected.