Kinetic approach to the Gaussian thermostat in a dilute sheared gas in the thermodynamic limit

A dilute gas of particles with short range interactions is considered in a shearing stationary state. A Gaussian thermostat keeps the total kinetic energy constant. For infinitely many particles it is shown that the thermostat becomes a friction force with constant friction coefficient. For finite number of particles N , the fluctuations around this constant are of order 1/ √ N , and distributed approximately Gaussian with deviations for large fluctuations. These deviations prohibit a derivation of fluctuation-dissipation relations far from equilibrium, based on the Fluctuation Theorem. PACS numbers: 05.20.Dd , 05.40.+j , 05.45.+b Typeset using REVTEX email: [email protected] 1 The interest in the relation between non-equilibrium statistical mechanics and microscopic equations of motion, which already occupied Boltzmann, has revived in recent years, on the one hand due to the development of chaos theory, but even more due to results from non-equilibrium molecular dynamics [1,2]. The main focus in the field is on stationary states. A stationary state, if it is not the equilibrium state, is the result of an external driving force. But this force performs work on the system, so it heats up (viscous heating, Ohmic heating). In simulations this is often remedied by the introduction of a mechanical thermostat: one adds a friction force, −α~vi, in the equation of motion for the velocity ~vi of each particle i, to keep the energy constant. For the thermostat variable α there are several choices. One could take it constant, but then one only gets a constant energy on average. It is also possible to have α time dependent, such that the total kinetic energy is constant (iso-kinetic Gaussian thermostat) or the total energy is constant (iso-energetic Gaussian thermostat) [2]. Neither of these thermostats are very realistic, as the dissipation of the heating would more likely occur at the boundary, where the system is in contact with a heat bath, say. Other boundary formulations where the driving force and the thermostat are combined have also been studied [3,4]. One hopes the choice of the thermostat doesn’t matter in the thermodynamic limit. The equivalence of a constant α thermostat, the iso-kinetic thermostat and iso-energetic thermostat was proposed by Gallavotti [5]. The extra term in the equations of motion destroys the Liouvillian character of the flow: a given volume in phase space will not retain that volume. As the available phase space is usually finite, this means that on average over time the volume either stays constant (conservative case) or contracts (dissipative case). In a dissipative system a stationary state can exist only on a course grained scale, the dissipation continues forever but on ever finer scales. This dissipation happens at a rate called the phase space contraction rate which is proportional to the average of the thermostat variable α. This rate can be identified [6,7] with the irreversible entropy production [8]. If we make this identification with a physical quantity, the precise implementation (iso-kinetic, iso-energetic, constant α, . . .) of the thermostat should not influence the average of α in the thermodynamic limit. Cohen [9] suggested that a mechanical and a physical thermostat may give the same results as long as the rate of heating is much less than the rate at which heat can be transported to the wall and absorbed there. This suggests that when the rate of heating becomes to large, the thermostat does make a difference. At that point one also expects the assumption of local equilibrium underlying non-equilibrium thermodynamics to break down, and the entropy production may no longer have the form that was used to identify it with α. We want to know which thermostat to use for analytic treatment of dilute gases in nonequilibrium stationary states. As we are interested in the limit of many particles, having to use an α dependent on all these particles would certainly make work more difficult. In other analytic work on non-equilibrium stationary states, one simply takes a constant α [10,11]. In this paper, the equivalence of an iso-kinetic Gaussian thermostat and a constant thermostat in the thermodynamic limit is demonstrated on the Boltzmann level (i.e. at low densities) for a sheared system of short range interacting particles.