The Einstein-Boltzmann Relation for Thermodynamic and Hydrodynamic Fluctuations

When making the connection between the thermodynamics of irreversible processes and the theory of stochastic processes through the fluctuation-dissipation theorem, it is necessary to invoke a postulate of the Einstein-Boltzmann type. For convective processes hydrodynamic fluctuations must be included, the velocity is a dynamical variable and although the entropy cannot depend directly on the velocity, $\delta^{2} S$ will depend on velocity variations. Some authors do not include velocity variations in $\delta^{2} S$, and so have to introduce a non-thermodynamic function which replaces the entropy and does depend on the velocity. At first sight, it seems that the introduction of such a function requires a generalisation of the Einstein-Boltzmann relation to be invoked. We review the reason why it is not necessary to introduce such a function, and therefore why there is no need to generalise the Einstein-Boltzmann relation in this way. We then obtain the fluctuation-dissipation theorem which shows some differences as compared with the non-convective case. We also show that $\delta^{2} S$ is a Liapunov function when it includes velocity fluctuations.Comment: 13 Page

New tests and applications of the worldline path integral in the first order formalism

We present different non-perturbative calculations within the context of Migdal's representation for the propagator and effective action of quantum particles. We first calculate the exact propagators and effective actions for Dirac, scalar and Proca fields in the presence of constant electromagnetic fields, for an even-dimensional spacetime. Then we derive the propagator for a charged scalar field in a spacelike vortex (i.e., instanton) background, in a long-distance expansion, and the exact propagator for a massless Dirac field in 1+1 dimensions in an arbitrary background. Finally, we present an interpretation of the chiral anomaly in the present context, finding a condition that the paths must fulfil in order to have a non-vanishing anomaly.Comment: 26 page

Dynamical phase coexistence: A simple solution to the "savanna problem"

We introduce the concept of 'dynamical phase coexistence' to provide a simple solution for a long-standing problem in theoretical ecology, the so-called "savanna problem". The challenge is to understand why in savanna ecosystems trees and grasses coexist in a robust way with large spatio-temporal variability. We propose a simple model, a variant of the Contact Process (CP), which includes two key extra features: varying external (environmental/rainfall) conditions and tree age. The system fluctuates locally between a woodland and a grassland phase, corresponding to the active and absorbing phases of the underlying pure contact process. This leads to a highly variable stable phase characterized by patches of the woodland and grassland phases coexisting dynamically. We show that the mean time to tree extinction under this model increases as a power-law of system size and can be of the order of 10,000,000 years in even moderately sized savannas. Finally, we demonstrate that while local interactions among trees may influence tree spatial distribution and the order of the transition between woodland and grassland phases, they do not affect dynamical coexistence. We expect dynamical coexistence to be relevant in other contexts in physics, biology or the social sciences.Comment: 8 pages, 7 figures. Accepted for publication in Journal of Theoretical Biolog