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Deterministic thermostats are frequently employed in non-equilibriummolecular dynamics simulations in order to remove the heat producedirreversibly over the course of such simulations. The simplest thermostat isthe Gaussian thermostat, which satisfies Gauss's principle of least constraintand fixes the peculiar kinetic energy. There are of course infinitely many waysto thermostat systems, e.g. by fixing $\sum\limits_i{|{p_i}|^{\mu + 1}}$. Inthe present paper we provide, for the first time, convincing arguments as towhy the conventional Gaussian isokinetic thermostat ($\mu=1$) is unique in thisclass. We show that this thermostat minimizes the phase space compression andis the only thermostat for which the conjugate pairing rule (CPR) holds.Moreover it is shown that for finite sized systems in the absence of an applieddissipative field, all other thermostats ($\mu=1$) perform work on the systemin the same manner as a dissipative field while simultaneously removing thedissipative heat so generated. All other thermostats ($\mu=1$) are thusauto-dissipative. Among all $\mu$-thermostats, only the $\mu=1$ Gaussianthermostat permits an equilibrium state.Comment: 27 pages including 10 figures; submitted for publication Journal of  Chemical Physic

New observations regarding deterministic, time-reversible thermostats and Gauss’s principle of least constraint