Applications of Dynamical Systems to Statistical Physics, Geometry, and Population Biology/Demography

ABSTRACT

Weiss

This proposal is to study several applications of dynamical systems to

statistical physics, geometry, and population biology/ecology. (i) The pressure and free energy are the two fundamental objects of study in the statistical physics of lattice spin systems. However, even for the simplest lattice spin systems, the information about the microscopic potential that the free energy captures is subtle and poorly understood. The PI has started a program to study whether, or to what extent, natural classes of Holder continuous potentials for certain one-dimensional lattice spin systems are determined by their free energy. We also plan to investigate striking similarities between the rigidity of free energy and fascinating rigidity problems in spectral geometry and number theory. (ii) Little is known about the

dynamics of the geodesic flow on positively curved manifolds. The PI

plans to continue studying the relations between positive curvature and

complicated dynamics of the geodesic flow. (iii) The PI has started a

program to systematically study the global dynamics and bifurcations for

nonlinear Leslie models where the fertility rates and survival

probabilities have various natural functional forms as functions of the

population size.

(i) The pressure and free energy are the two fundamental objects of study in the statistical physics of lattice spin systems. Lattice spin systems provide an important and illuminating family of models in statistical physics, condensed matter physics, and chemistry. For instance, phase transitions correspond to non-differentiability for some derivative of free energy. However, even for the simplest lattice spin systems, the information about the microscopic potential that the free energy captures is subtle and poorly understood. The PI has started a program to study whether, or to what extent, potentials for

one-dimensional lattice systems are determined by their free energy. We

hope this work will provide new insights into this important, yet

mysterious, quantity. (ii) Essentially all demographic and animal

population models in current use are based on the linear Leslie model.

Many population biologists, ecologists, and demographers are now looking

to nonlinear population models for more accurate population forecasting.

The PI has started a program to systematically study the global dynamics

and bifurcations for nonlinear Leslie models where the fertility rates and

survival probabilities have various natural functional forms as functions

of the population size. One of our ultimate goals is to create a

``population modeling toolbox'' which could be used by a wide range of

population modelers to more accurately predict animal populations.