Enforcing Detailed Balance in the Borgnakke-Larsen Redistribution Method

It is often observed that Direct Simulation Monte Carlo (DSMC) simulations either fail or succeed to reach and maintain equilibrium depending on the choice of temperature dependent relaxation models for internal energy modes, with no clear consensus in the literature as to why. This work alleviates the ambiguity regarding the root cause of these failures by presenting a rigorous theoretical derivation of the requirement for satisfying detailed balance within the Borgnakke-Larsen method for energy redistribution, which is often used to implement relaxation models within DSMC. Specifically, it is shown that the Borgnakke-Larsen method maintains detailed balance if and only if the probability of internal-energy exchange during a collision depends only on collision invariants (e.g., total energy). The consequences of this result are explored in the context of several published relaxation models and definitions of relaxation temperature. The developed theory can be used when implementing existing or new relaxation models and will ensure detailed balance is satisfied.