Kinetic Equations for Unreactive Processes

Kinetic equations for the distribution functions of non-reacting gases are, in fact, a generalization of the Boltzmann equation, i.e. equations for fluxes of species incoming to and outgoing from certain quantum states as a result of molecular collisions. 

Consider, for instance, the energy exchange between molecules A and B of a two-component gas mixture. Let a and bj be the populations of the quantum states i and j of these molecules normalized to the total number of molecules per unit volume 

Here i and j stand for internal quantum numbers of colliding molecules and for both the values and directions of molecular velocities. If k m denotes the rate coefficient for the elementary process of transition from the initial state ij to the final state Im in one collision between two A molecules (k im and k ni stand for rate constants in collisions of B — B and A — B, respectively), then the rate of change in population a is equal to the difference in fluxes to and from this state 

The two first rhs terms allow for transitions from states i upon collisions of A with A and B (summation over j) to all final states of both colliding molecules (summation over 1 and m). 

The two last rhs terms take into account the transition to state i from all possible states of the collising pairs A and B. The rate constants k m k are expressed by 

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