Deterministic dynamics

The Boltzman Machine generalizes the Hopfield model in two ways (1) like the simple stochastic variant discussed above, it t(>o substitutes a stochastic update rule for Hopfield s deterministic dynamics, and (2) it separates the neurons in the net into sets of visible and hidden units. Figure 10.8 shows a Boltzman Machine in which the visible neurons have been further subdividetl into input and output sets. 

As a first step toward a TST treatment of the stochastically driven dynamics, it is crucial to assume, just as in the autonomous case, that the deterministic dynamics has a fixed point that marks the location of an energetic barrier between reactants and products. In the case of Eq. (13), the fixed point is given by a saddle point q0 of the potential U(q). The reaction rate is determined by the... 

The linearization of the deterministic dynamics allows one to solve the equations of motion explicitly. Equation (15) can be rewritten as a first-order equation of motion in the 2/V-dimensional phase space with the coordinates... 

For deterministic dynamics the state zt+At at time t + At is of course completely determined by the state of the system zt a time step At earlier. Therefore, the single-time-step transition probability p(zt -> zt+At) can be written in terms of a delta function... 

The authors then ask the following question Do there exist deterministic dynamical systems that are, in a precise sense, equivalent to a monotonous Markov process The question can be reformulated in a more operational way as follows Does there exist a similarity transformation A which, when applied to a distribution function p, solution of the Liouville equation, transforms the latter into a function p that can also be interpreted as a distribution function (probability density) and whose evolution is governed by a monotonous Markov process An affirmative answer to this question requires the following conditions on A (MFC) ... 

MSN.97. B. Misra, 1. Prigogine, and M. Courbage, From deterministic dynamics to probabilistic descriptions, Proc. Natl. Acad. Sci. USA 76, 3607-3611 (1979). 

Dynamic aibcation is an alternative procedure in which the allocation of treatment to a subject is influenced by the current balance of allocated treatments and, in a stratified trial, by the stratum to which the subject belongs and the balance within that stratum. Deterministic dynamic allocation procedures should be avoided and an appropriate element of randomisation should be incorporated for each treatment allocation. ... 

Stochastic approximations such as random walk or molecular chaos, which treat the motion as a succession of simple one- or two-body events, neglecting the correlations between these events implied by the overall deterministic dynamics. The analytical theory of gases, for example, is based on the molecular chaos assumption, i.e. the neglect of correlations betweeen consecutive collision partners of the same molecule. Another example is the random walk theory of diffusion in solids, which neglects the dynamical correlations between consecutive jumps of a diffusing lattice vacancy or interstitial. 

We emphasize that the question of stability of a CA under small random perturbations is in itself an important unsolved problem in the theory of fluctuations [92-94] and the difficulties in solving it are similar to those mentioned above. Thus it is unclear at first glance how an analogy between these two unsolved problems could be of any help. However, as already noted above, the new method for statistical analysis of fluctuational trajectories [60,62,95,112] based on the prehistory probability distribution allows direct experimental insight into the almost deterministic dynamics of fluctuations in the limit of small noise intensity. Using this techique, it turns out to be possible to verify experimentally the existence of a unique solution, to identify the boundary condition on a CA, and to find an accurate approximation of the optimal control function. 

We note that for p =0 and /u 0 the dynamics of (37) reduces to the deterministic dynamics of the original system (35) in the absence of control (u(t) = 0). So we begin our analysis by considering some relevant properties of the deterministic dynamics of a periodically driven nonlinear oscillator. 

Deterministic dynamics of biochemical reaction systems can be visualized as the trajectory of (ci(t), c2(t), , c v(0) in a space of concentrations, where d(t) is the concentration of ith species changing with time. This mental picture of path traced out in the N-dimensional concentration space by deterministic systems may prove a useful reference when we deal with stochastic chemical dynamics. In stochastic systems, one no longer thinks in terms of definite concentrations at time t rather, one deals with the probability of the concentrations being xu x2, , Wy at time t ... 

Although the transient FR (1.1) is usually applied to systems that are initially at equilibrium, the consideration of the derivation shows that this is not necessary, and the rmderlying requirement is just that the initial distribution is known. For many-particle deterministic dynamics, the nonequilibrium distribution function is... 

For deterministic systems, the FR takes on the form (1.1) and the dissipation function is given by (2.1). However, in the case of stochastic dynamics, the same process might be modelled at different levels with different dynamics, and for each model a different fluctuation relation may be obtained. Therefore there are more papers on stochastic systems than on deterministic dynamics, as derivations for new dynamics allow new systems to be treated. This is particularly true for the Jarzynski relation, as discussed in section 3.2. 

Recently, there has been interest in considering FR for systems coupled to heat reservoirs at different temperatures." The relevant FR for systems modelled using deterministic dynamics was determined and tested numerically in 2001, however interest in heat flow for simple models is important for studies aimed at determining the nonequilibrium temperatures and studying Fourier s Law, and therefore consideration of FR in these systems has received some attention. Gomez-Marin and Sancho and were unable to verify an FR proposed by Jarz5mski and Wojcik, presumably due to neglect of boundary terms in the theoretical analysis that were relevant for their numerical calculations. Visco" has studied the GC FR in a model heat flow problem. 

Due to the changes in the dynamics, a general relationship for stochastic dynamics is not available like it is for deterministic dynamics. However, for mesoscopic systems, a mesoscopic FR is useful. Therefore, there has been much work on developing stochastic models with different conditions. Andrieux and Gaspard developed a stochastic fluctuation relation for nonequilibrium systems whose dynamics can be described by Schnakenberg s network theory (e.g. mesoscopic electron transport, biophysical models of ion transport and some chemical reactions). Due to early experimental work on protein unfolding and related molecular motors, and their ready treatment by stochastic dynamics, a number of papers have appeared that model these systems and test the or JE for these. FR... 

Elementary processes in chemical dynamics are universally important, besides their own virtues, in that they can link statistical mechanics to deterministic dynamics based on quantum and classical mechanics. The linear surprisal is one of the most outstanding discoveries in this aspect (we only refer to review articles [2-7]), the theoretical foundation of which is not yet well established. In view of our findings in the previous section, it is worth studying a possible origin of the linear surprisal theory in terms of variational statistical theory for microcanonical ensemble. 

S. M. Ulam and J. von Neumann, a combination of stochastic and deterministic dynamics ,... 

The deterministic dynamics of this cluster model has been investigated in [32, 33]. In addition to IP3 mediated Ca liberation, we considered sarco-endoplasmic reticulum calcium ATPase (SERCA) pumps, which transport Ca from the cytosol to the ER, and a leak flux. The stationary Ca concentration profile that results from these three fluxes is... 

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