By Max Born, H. S. Green

This paper outlines a basic thought whose item is to supply a foundation from which all of the equilibrium and dynamical houses of beverages will be investigated. a suite of multiform distribution capabilities is outlined, and the generalized continuity equations chuffed via those features are derived. through introducing the equations of movement, a suite of kinfolk is got from which the distribution features can be made up our minds. it really is proven that Boltzmann's equation within the kinetic concept of gases follows as a specific case, and that, in equilibrium stipulations, the speculation offers effects in step with statistical mechanics. An critical equation for the radial distribution functionality is got that's the normal generalization of 1 received by way of Kirkwood for 'rigid round molecules'. ultimately, it's indicated how the idea should be utilized to unravel either equilibrium and dynamical difficulties of the liquid nation.

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1 Reaction–Diﬀusion Systems with Two Intermediates 47 which is zero at the stable stationary state; and similarly the ﬁrst derivative with respect to Yi . 34) that is larger than zero, and hence Φ is a minimum at the stable stationary state. The derivative with respect to time is d Φ[{Xi , Yi }] = dt i dXi {µ(Xi ) − µ(XiS )} dt + dYi {µ(Yi ) − µ(YiS )} . 34) is negative semideﬁnite, so that the system tends towards the minimum of Φ, that is towards the stable stationary state. Thus the function Φ is a Liapunov function of the system.

16) for the ﬂuctuational trajectory starting at the nth stable stationary state xsn with p = 0 at t = −∞ and ending at x at t = 0. 17) run in an apparatus as in Fig. 1 of Chap. 1, with the pressures of A and B held constant. 18) and dY dt det = k3 X − (k4 + k5 )Y + k6 B = −[(k4 + k5 )(Y − Ys )] − k3 (X − Xs )]. 1. 1, taken from [1]. Now we use the Hamiltonian equations of motion to obtain the ﬂuctuational trajectories: dX dt dY dt ﬂ ﬂ = k1 A − exp(px ) − k2 X exp(−px ) − k3 X exp(py − px ) +k4 Y exp(px − py ), = k3 X − exp(py − px ) − k4 Y exp(px − py ) −k5 Y exp(−py ) + k6 B exp(py ), dpx = k2 [1 − exp(−px )] + k3 [1 − exp(py − px )], dt dpy = k4 [1 − exp(px − py )] + k5 [1 − exp(−py )].

1] . ˆ + (x, p H ˆ) . . 9) r Thus we have formally, and exactly, converted the master equation to a Schroedinger equation. This has the substantial advantage that we can apply well-known approximations in quantum mechanics to obtain solutions to the master equation. B. approximation valid for semiclassical cases, those for which Planck’s constant formally approaches zero. 8) is that of large volumes (large numbers of particles). 10) where Sn will be shown to be the classical action of a ﬂuctuational trajectory accessible from the nth stable stationary state.

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