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A field theory approach to the dynamics of classical particles

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## A field theory approach to the dynamics of classical particles

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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