Transcript A field theory approach to the dynamics of classical particles

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
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Motivation
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Theory
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How can we investigate ergodic-nonergodic transitions?
What do we need in a theory of dense fluids?
What does our self-consistent theory look like?
Results
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What does our theory say about ergodic-nonergodic transitons?
Can we derive a mode-coupling theory-like kinetic equation and
memory function?
Motivation
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Why study dense fluids?
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What are the shortcomings in the current theory (MCT)?
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Interested in long-time behavior
Want to investigate ergodic-nonergodic transitions
An ad hoc construction
An approximation without a clear method for corrections
What do we really want in our theory?
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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
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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
(1)Off-diagonal components give rise to self-consistent
statics
(2)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
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Demonstrated a theory for treating dense fluids
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Field theory-based
Self-consistent
Perturbative control
Able to study both statics and dynamics
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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
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Smoluchowski Dynamics
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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
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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,
Travel Funding
NSF-MRSEC (UChicago)
James Franck Institute
(UChicago