Zeroth Law investigation on the logarithmic thermostat

The Zeroth Law implies that the three systems, each separately in equilibrium and having the same temperature, must remain so when brought in pairwise or simultaneous thermal contact with each other. We examine numerically the conformity of the logarithmic thermostat with the Zeroth Law of thermodynamics. Three specific scenarios, with different heat reservoirs, are investigated. For each scenario, the system of interest, S₁ - a single harmonic oscillator, is coupled with two heat reservoirs, S₂ and S₃. S₂ and S₃ are variously chosen to be from the Nosé-Hoover, the Hoover-Holian, the C_1,2 and the logarithmic thermostats. In the scenarios involving logarithmic thermostat, we observe a violation of the Zeroth Law of thermodynamics, in computationally achievable time, at low to moderate coupling strengths: (i) the kinetic and configurational temperatures of the systems are different, (ii) momentum distribution of log thermostat is non-Gaussian, and (iii) a temperature gradient is created between the kinetic and configurational variables of the log thermostat.