Very fast simulated re-annealing

Simulated annealing: Practice versus theory

Although thermodynamic fluctuation theory originated from statistical mechanics, it may be put on a completely thermodynamic basis, in no essential need of any microscopic foundation. This review views the theory from the macroscopic perspective, emphasizing, in particular, notions of covariance and consistency, expressed naturally using the language of Riemannian geometry. Coupled with these concepts is an extension of the basic structure of thermodynamic fluctuation theory beyond the classical one of a subsystem in contact with an infinite uniform reservoir. Used here is a hierarchy of concentric subsystems, each of which samples only the thermodynamic state of the subsystem immediately larger than it. The result is a covariant thermodynamic fluctuation theory which is plausible beyond the standard second-order entropy expansion. It includes the conservation laws and is mathematically consistent when applied to fluctuations inside subsystems. Tests on known models show improvements. Perhaps most significantly, the covariant theory offers a qualitatively new tool for the study of fluctuation phenomena: the Riemannian thermodynamic curvature. The thermodynamic curvature gives, for any given thermodynamic state, a lower bound for the length scale where the classical thermodynamic fluctuation theory based on a uniform environment could conceivably hold. Straightforward computation near the critical point reveals that the curvature equals the correlation volume, a physically appealing finding. The combination of the interpretation of curvature with a well-known proportionality between the free energy and the inverse of the correlation volume yields a purely thermodynamic theory of the critical point. The scaled equation of state follows from the values of the critical exponents. The thermodynamic Riemannian metric may be put into the broader context of information theory.

Riemannian geometry in thermodynamic fluctuation theory

https://escholarship.org/content/qt8p6601jj/qt8p6601jj.pdf?t=p0hgz0

The semiclassical evolution of wave packets

A general conceptual framework for large-scale neocortical dynamics based on data from many laboratories is applied to a va- riety of experimental designs, spatial scales, and brain states. Partly distinct, but interacting local processes (e.g., neural networks) arise from functional segregation. Global processes arise from functional integration and can facilitate (top down) synchronous activity in re- mote cell groups that function simultaneously at several different spatial scales. Simultaneous local processes may help drive (bottom up) macroscopic global dynamics observed with electroencephalography (EEG) or magnetoencephalography (MEG). A local/global dynamic theory that is consistent with EEG data and the proposed conceptual framework is outlined. This theory is neutral about properties of neural networks embedded in macroscopic fields, but its global component makes several qualitative and semiquantitative predictions about EEG measures of traveling and standing wave phenomena. A more general "metatheory" suggests what large-scale quantitative theories of neocortical dynamics may be like when more accurate treatment of local and nonlinear effects is achieved. The theory describes the dynamics of excitatory and inhibitory synaptic action fields. EEG and MEG provide large-scale estimates of modulation of these synaptic fields around background levels. Brain states are determined by neuromodulatory control parameters. Purely local states are dominated by local feedback gains and rise and decay times of postsynaptic potentials. Dominant local frequen- cies vary with brain region. Other states are purely global, with moderate to high coherence over large distances. Multiple global mode frequencies arise from a combination of delays in corticocortical axons and neocortical boundary conditions. Global frequencies are iden- tical in all cortical regions, but most states involve dynamic interactions between local networks and the global system. EEG frequencies may involve a "matching" of local resonant frequencies with one or more of the many, closely spaced global frequencies.

/pdf/toward-a-quantitative-description-of-large-scale-neocortical-268lujxhjg.pdf

Toward a quantitative description of large-scale neocortical dynamic function and EEG.

I: Functional integrals defined as limits of discretized expressions.- II: Correspondence rules and functional integral representations.- III: Functional integral representations of expectation values. Time-ordered products.- IV: Perturbation expansions.- V: Short time propagators and the relations between them.- VI: Covariant definitions of functional integrals.- VII: Functional integral methods in Fokker-Planck dynamics.- VIII: Product integrals.- IX: The semiclassical expansion in phase space.- X: The semiclassical expansion in configuration space.- XI: Other approaches.- XII: Computation of the propagator on the sphere S3.- References.

Functional Integration and Semiclassical Expansions

We study functional integrals in phase space and show that the definition without limiting procedure by a formal series expansion has the same ambiguities as the definition through discretization, these last ones being related to ordering problems. We exhibit the way to obtain the prescription that makes the series expansion unambiguous and study the mechanism that makes the expansion independent of the chosen discretization.

Discretization Problems of Functional Integrals in Phase Space

We study a method proposed recently that gives a unified view of the properties of stochastic fields determined by Langevin-type equations. The functional integrals involved are shown to need an additional prescription to be defined, this prescription being related to the relative order of noncommuting operators in the corresponding operator formalism which we characterize in detail. It is proved that the prescription is responsible for an additional term which has given rise to difficulties of interpretation in the Onsager-Machlup path probability density. We prove that all results are independent of this term. In particular in perturbation theory the cancellation mechanism of the prescription dependence is studied in detail.

Functional integral methods for stochastic fields

We give a precise definition of the functional integrals involved by means of different discretization prescriptions. The value of the functional integral is prescription dependent and this leads to a variety of representations with different Lagrangians. We discuss general covariance and we derive in a unified way, all the lagrangians proposed in the literature together with their associated discretizations. We make also a critical examination of previous results.

Functional integrals and the Fokker-Planck equation

A short derivation of the Feynman Lagrangian for general diffusion processes is given by a technique relying on the use of different discretisations which are related by equivalence relations under the n-dimensional integral whose limit is the path integral. In this way calculation of the differential equation satisfied by the path integral is avoided.

http://iopscience.iop.org/article/10.1088/0305-4470/13/2/013/pdf

F. Langouche

Papers

Functional Integration and Semiclassical Expansions

Discretization Problems of Functional Integrals in Phase Space

Functional integral methods for stochastic fields

Functional integrals and the Fokker-Planck equation

Short derivation of feynman lagrangian for general diffusion processes