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A field theory approach to the dynamics of classical particles David D. McCowan with Gene F. Mazenko and Paul Spyridis The James Franck Institute and the Department of Physics
Outline • Motivation • How can we investigate ergodic-nonergodic transitions? • What do we need in a theory of dense fluids? • Theory • What does our self-consistent theory look like? • Results • What does our theory say about ergodic-nonergodic transitons? • Can we derive a mode-coupling theory-like kinetic equation and memory function?
Motivation • Why study dense fluids? • Interested in long-time behavior • Want to investigate ergodic-nonergodic transitions • What are the shortcomings in the current theory (MCT)? • An ad hoc construction • An approximation without a clear method for corrections • What do we really want in our theory? • Developed from first principles • A clear prescription for corrections • Self-consistent perturbative development
Theory – Setup For concreteness, we will treat Smoluchowski (dissipative) dynamics and begin with a Langevin equation for the coordinate Ri where the force is due to a pair potential and the noise is Gaussian distributed But we want to build up a field theory formalism • Create a Martin-Siggia-Rose action, with the coordinate and conjugate response as our variables
Theory – Generating Functional Our generating function is of the form Leads us to define our fields as (density) (response)
Theory – Cumulants The generating functional can be used to form cumulants and the components are given by For example: (density-density) (response-response) (density-response) (FDT)
Results – Perturbation Expansion • Vertex functions are defined via Dyson’s equation • and we may make perturbative approximations to • Off-diagonal components give rise to self-consistent statics • Diagonal components give rise to the kinetics of the typical (MCT) form
Results – Statics/Pseudopotential At lowest nontrivial order, we have which we can place into the static structure factor and self-consistently solve for the potential This in turn yields the average density
Results – Kinetic Equation At lowest nontrivial order, we have and this can be used in our derived kinetic equation We find characteristic slowing down at large densities and we observe an ergodic-nonergodic transition at a value of η = 0.76 for Percus-Yevick hard spheres
Conclusion • Demonstrated a theory for treating dense fluids • Field theory-based • Self-consistent • Perturbative control • Able to study both statics and dynamics • Has a clear mechanism for investigating ergodic-nonergodic transitions • Capable of generating MCT-like kinetic equation and memory function • Gives a drastic slowing-down and three step decay in the dynamics at high density
References • Smoluchowski Dynamics • G. F. Mazenko, D. D. McCowan and P. Spyridis, "Kinetic equations governing Smoluchowski dynamics in equilibrium," arXiv:1112.4095v1 (2011). • G. F. Mazenko, "Smoluchowski dynamics and the ergodic-nonergodic transition," Phys Rev E 83 041125 (2011). • G. F. Mazenko, "Fundamental theory of statistical particle dynamics," Phys Rev E 81 061102 (2010). • Newtonian Dynamics • S. P. Das and G. F. Mazenko, “Field Theoretic Formulation of Kinetic theory: I. Basic Development,” arXiv:1111.0571v1 (2011). • Research Funding • Department of Physics, UChicago • Joint Theory Institute, UChicago • Travel Funding • NSF-MRSEC (UChicago) • James Franck Institute (UChicago