Exact Time-Dependent Thermodynamic Relationships for a Brownian Particle Navigating Complex Networks

Abstract

The thermodynamic characteristics of systems driven out of equilibrium are examined for M Brownian ratchets organized in a complex network. The precise time-dependent solution reveals that the entropy S, entropy production ep(t), and entropy extraction hd(t) of the system in complex networks increase with system size, which is plausible as these thermodynamic quantities display extensive properties. In other words, as the number of lattice sites increases, the entropy S, entropy production ep(t), and entropy extraction hd(t) increase, demonstrating that these complex networks cannot be reduced to the corresponding one-dimensional lattice. Conversely, the rates for thermodynamic quantities such as velocity V, entropy production rate e˙p(t), and entropy extraction rate h˙ d(t) become independent of the network size in the long-term limit. The exact analytic results also indicate that the free energy decreases with system size. The model system is further analyzed by incorporating heat transfer via kinetic energy. Since heat exchange via kinetic energy does affect the energy extraction rate, the heat dumped to the cold reservoirs also contributes to internal entropy production. Consequently, such systems exhibit a higher degree of irreversibility. The thermodynamic features of a system operating between hot and cold baths are also compared and contrasted with a system functioning in a heat bath where temperature varies linearly along the reaction coordinate. Regardless of the network arrangements, the entropy, entropy production, and extraction rates are significantly larger for the linearly varying temperature case than for a system operating between hot and cold baths.