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The stationary nonequilibrium Gibbsian ensemble representing a harmonic crystal in contact with several idealized heat reservoirs at different temperatures is shown to have a Gaussian Γ space distribution for the case where the stochastic interaction between the system and heat reservoirs may be represented by Fokker-Planck-type operators. The covariance matrix of this Gaussian is found explicitly for a linear chain with nearest-neighbor forces in contact at its ends with heat reservoirs at temperatures T1 and TN, N being the number of oscillators. We also find explicitly the covariance matrix, but not the distribution, for the case where the interaction between the system and the reservoirs is represented by very ``hard'' collisions. This matrix differs from that for the previous case only by a trivial factor. The heat flux in the stationary state is found, as expected, to be proportional to the temperature difference (T1 − TN) rather than to the temperature gradient (T1 − TN)/N. The kinetic temperature of the jth oscillator T(j) behaves, however, in an unexpected fashion. T(j) is essentially constant in the interior of the chain decreasing exponentially in the direction of the hotter reservoir rising only at the end oscillator in contact with that reservoir (with corresponding behavior at the other end of the chain). No explanation is offered for this paradoxical result.
Rieder et al. (1967) studied this question.