By Stuart A. Rice
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Additional info for Advances in Chemical Physics, Vol. 140
If the matrix is taken to be antisymmetric, ÈðxÞ ¼ ÀÈðxÞT ð123Þ then the quasi-orthogonality condition (121) is automatically satisﬁed. That same condition shows that ÈðxÞ ¼ E ÈðExÞT E ð124Þ since f 1 ðExÞ ¼ ÀEf 1 ðxÞ. The derivative of the second entropy, Eq. (109), is qSð2Þ ðx0 ; xjtÞ ¼ ^tg ðxÞ x þ12½XðxÞ þ 2^tf 1 ðxÞ 0 0 qx ð125Þ Hence the most likely terminal position is xðx; tÞ ¼ x À jtjg ðxÞÀ1 ½12XðxÞ þ ^tf 1 ðxÞ 0 jtj ¼ x À g ðxÞÀ1 XðxÞ À tÈðxÞXðxÞ 2 0 x À jtjLðx; ^tÞXðxÞ ð126Þ 34 phil attard The antisymmetric part of nonlinear transport matrix is not uniquely deﬁned (due to the nonuniqueness of È).
86). The transition x0 ! x00 is determined by one-half of the external change in the total ﬁrst entropy. The factor of 12 occurs for the conditional transition probability with no speciﬁc correlation between the terminal states, as this preserves the singlet probability during the reservoir induced transition [4, 8, 80]. The implicit assumption underlying this is that the conductivity of the reservoirs is much greater than that of the subsystem. The second entropy for the stochastic transition is the same as in the linear case, Eq.
The quadratic term is negative and represents the entropy cost of ordering the system dynamically; whether the departure from zero is positive or negative, any ﬂuctuation in the ﬂux represents order and is unfavorable. The linear term can be positive or negative; it is this term that encourages a nonzero ﬂux that drives the system back toward equilibrium in the future, which obviously increases the ﬁrst entropy. ) Hence the optimization procedure corresponds to balancing these two terms, with the quadratic term that is unfavorable to dynamic order preventing large ﬂuxes, where it dominates, and the linear term increasing the ﬁrst entropy and dominating for small ﬂuxes.