A field theory approach to the dynamics of classical particles

1
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

2
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?

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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

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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

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Theory Generating Functional

• Our generating function is of the form
• Leads us to define our fields as
• (density) (response)

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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)

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Results Perturbation Expansion

• Vertex functions are defined via Dysons 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

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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
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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 ? 0.76 for Percus-Yevick hard
  spheres

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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

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References

• Smoluchowski Dynamics
• G. F. Mazenko, D. D. McCowan and P. Spyridis,
  "Kinetic equations governing Smoluchowski
  dynamics in equilibrium," arXiv1112.4095v1
  (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, arXiv1111.0571v1 (2011).

Research Funding Department of Physics,
UChicago Joint Theory Institute, UChicago
Travel Funding NSF-MRSEC (UChicago) James Franck
Institute (UChicago