By Harrison D.M.
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Extra resources for About Mass-Energy Equivalence
Prigogine, 1962), or(t) = exp( -iLt)or(O) for 8L/8t = o. L is hermitian with respect to the inner product (or, or') := oro~ dpdq (that is (or, LoH = (Lor, oH), as can again be shown by partial integration. As is known from the Schrodinger equation, this means that the Liouville equation conserves these inner products. 27) 44 3. lder)Ler as the time derivative. In particular, the norm IIerl1 2 = (er, er) = J e} dpdq = er corresponding to this inner product is constant in time. 3 The conservation of such measures under the Liouville equation confirms that the r -space volume is an appropriate measure for non-countable sets of states (Ehrenfest, 1911): the 'number' of states in an ensemble must not change under a deterministic dynamics.
In many cases such a weakly coupled environment may even co-determine macroscopic effects (thus causing an effective macroscopic indeterminism), as is much discussed in the theory of chaos. (See, for example, Schuster, 1984). The essence of these considerations is that macroscopic systems, aside from the whole universe, may never be considered as dynamically isolated even when they are thermodynamically closed, that is, when any exchange of heat with the environment is completely negligible. In quantum mechanics, this microscopic coupling to the environment will also have fundamental kinematical consequences (see Sect.
By this implicit 'renormalization of entropy' one adds to the infinite negative entropy of the exact state an infinite positive contribution that corresponds to the smoothing. The resulting 'representative ensembles' of states with finite measure therefore define probabilities in the sense mentioned in the introduction to this chapter. They possess finite Boltzmannian entropies which do not depend on the precise values of the smoothing widths over a wide range, provided the discrete distribution is already smooth in the mean.
About Mass-Energy Equivalence by Harrison D.M.