Analysis, et Cetera

Analysis, et Cetera

Research Papers Published in Honor of Jürgen Moser's 60th Birthday
1990, Pages 677-694
Analysis, et Cetera

On the Derivation of Conservation Laws for Stochastic Dynamics

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The problem of deriving the macroscopic conservation laws, or the Euler equations, in a rigorous fashion from the microscopic equations of classical mechanics is still largely open, even though the physics is very well understood. With suitable scaling of space and time, a general picture emerges. At the microscopic or grain level, particles are moving very rapidly. In fact, the motion is so rapid that it is impossible to follow the motion of individual particles in the prevailing time scale. But the conservation laws deal with the evolution of the spatial distribution of quantities that are conserved in encounters between particles at the microscopic level. They involve an exchange of sorts between particles, and as a result, the macroscopic distribution of these quantities undergoes only changes of order one in the prevailing time scale. Much of the motion of the particles is thermal motion, and only a small part acts coherently to produce macroscopic changes in the distribution of these conserved quantities.

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