By Derek F. Lawden
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Additional resources for A Course in Applied Mathematics, Vols 1 & 2
However, these really represent the effect of uncertainty in the initial conditions for the system and its environment: indeﬁniteness in some of these initial environmental conditions might only have an impact upon the system at a later time. For example, the uncertainty in the velocity of a particle in a gas increases as particles that were initially far away, and that were poorly speciﬁed at the initial time, have the opportunity to move closer and interact. The evolution equations are not time reversal symmetric since the principle of causality is assumed: the probability of a system conﬁguration depends upon events that precede it in time, and not on events in the future.
12) in the form ð 1 1 Mn ðv; tÞ ¼ ð1:20Þ dDvðDvÞn TðDvjv; tÞ ¼ hðvðt þ tÞ À vðtÞÞn i; t t and in the continuum limit where t ! 0. This requires an equivalence between the average of ðDvÞn over a transition probability density T and the average over the statistics of the noise j. We integrate Eq. 19) for small t to get vðt þ tÞ À vðtÞ ¼ Àc ð tþt t vdt þ b ð tþt t jðt0 Þdt0 % ÀcvðtÞt þ b ð tþt jðt0 Þdt0 ; t ð1:21Þ and according to the properties of the noise, this gives hdvi ¼ Àcvt with dv ¼ vðt þ tÞ À vðtÞ, such that M1 ðvÞ ¼ h_vi ¼ Àcv.
Intuitively, we understand that we should be comparing the probability of observing some forward and reverse behavior, but these ideas need to be made concrete. Let us proceed in a manner that allows us to make a more direct connection between irreversibility and our consideration of Loschmidt’s paradox. First, let us imagine a system that evolves under some suitable stochastic dynamics. We speciﬁcally consider a realization or trajectory that runs from time t ¼ 0 to t ¼ t. Throughout this process, we imagine that any number of system parameters may be subject to change.
A Course in Applied Mathematics, Vols 1 & 2 by Derek F. Lawden