Multiple phases in stochastic dynamics: geometry and probabilities

Stochastic dynamics is generated by a matrix of transition probabilities. Certain eigenvectors of this matrix provide observables, and when these are plotted in the appropriate multi-dimensional space the phases (in the sense of phase transitions) of the underlying system become manifest as extremal points. This geometrical construction, which we call an \textit{observable-representation of state space}, can allow hierarchical structure to be observed. It also provides a method for the calculation of the probability that an initial points ends in one or another asymptotic state

Ratcheting up energy by means of measurement

The destruction of quantum coherence can pump energy into a system. For our examples this is paradoxical since the destroyed correlations are ordinarily considered negligible. Mathematically the explanation is straightforward and physically one can identify the degrees of freedom supplying this energy. Nevertheless, the energy input can be calculated without specific reference to those degrees of freedom.Comment: To appear in Phys. Rev. Let

Path integral in a magnetic field using the Trotter product formula

The derivation of the Feynman path integral based on the Trotter product formula is extended to the case where the system is in a magnetic field.Comment: To appear in the American Journal of Physics, 200

Imaging geometry through dynamics: the observable representation

For many stochastic processes there is an underlying coordinate space, $V$, with the process moving from point to point in $V$ or on variables (such as spin configurations) defined with respect to $V$. There is a matrix of transition probabilities (whether between points in $V$ or between variables defined on $V$) and we focus on its ``slow'' eigenvectors, those with eigenvalues closest to that of the stationary eigenvector. These eigenvectors are the ``observables,'' and they can be used to recover geometrical features of $V$

A Kinetic Model of the Muscular Contraction

In this article, we build a self consistent mean field deterministic model for the muscular contraction. The two main variables are the number of free myosin heads and the number of myosin heads attached to the actin, just after attachment. The model is natural in the sense that it respects the physico-chemical natural constraints. We calculate the stationary state, prove that it is stable and calculate the efficiency

Slow relaxation, confinement, and solitons

Millisecond crystal relaxation has been used to explain anomalous decay in doped alkali halides. We attribute this slowness to Fermi-Pasta-Ulam solitons. Our model exhibits confinement of mechanical energy released by excitation. Extending the model to long times is justified by its relation to solitons, excitations previously proposed to occur in alkali halides. Soliton damping and observation are also discussed

Topological methods for searching barriers and reaction paths

We present a family of algorithms for the fast determination of reaction paths and barriers in phase space and the computation of the corresponding rates. The method requires the reaction times be large compared to the microscopic time, irrespective of the origin - energetic, entropic, cooperative - of the timescale separation. It lends itself to temperature cycling as in simulated annealing and to activation-relaxation routines. The dynamics is ultimately based on supersymmetry methods used years ago to derive Morse theory. Thus, the formalism automatically incorporates all relevant topological information.Comment: 4 pages, 4 figures, RevTex

Nonlinear Dirac and diffusion equations in 1 + 1 dimensions from stochastic considerations

We generalize the method of obtaining the fundamental linear partial differential equations such as the diffusion and Schrodinger equation, Dirac and telegrapher's equation from a simple stochastic consideration to arrive at certain nonlinear form of these equations. The group classification through one parameter group of transformation for two of these equations is also carried out.Comment: 18 pages, Latex file, some equations corrected and group analysis in one more case adde