R Mike L Evans
Department of Applied Mathematics

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Mike Evans



Statistical Mechanics

My research interests are in theoretical physics, specifically non-equilibrium statistical mechanics, soft matter physics and the development of new methods of analysis. Non-equilibrium statistical mechanics is a very broad topic, with applications ranging from the flow of molten plastics to Darwinian evolution.
Soft matter includes colloids, emulsions, liquid crystals, amphiphiles and polymeric fluids. It's ubiquitous, constituting virtually all foods, biological materials, and pretty much anything squidgy. Such materials have useful properties that are exploited in industrial applications including plastics, displays, adhesives, coatings, cosmetics and countless others. But my interest is not motivated so much by their useful applications as by the elegant physics that governs their behaviour.
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In soft matter, we find an exotic menagerie of complex structures that can be understood in terms of their interesting symmetries and statistical properties. For instance, in this idealized simulation of amphiphiles (molecules with a water-loving end and a grease-loving end, like in washing-up liquid), the interplay between random thermal motion and the stickiness of the greasy ends of the molecules (here coloured black) leads to the spontaneous formation of worm-like structures called micelles, or striped structures called "lamellae" that form at higher chemical potential. These structures ultimately result from the statistics and sysmmetries of the molecules' random motion.

When soft matter flows, it exhibits much more interesting behaviour than simple fluids, including flow-alignment, shear-banding, jamming, and even (in the case of several polymer melts) expelling itself completely from the measurement apparatus.

Polydispersity
Equilibrium properties of such materials become difficult to calculate if the constituent particles are not all identical, but vary in size, shape, charge etc. I have developed methods for simplifying the mathematics associated with such 'polydisperse' substances, so that their 'phase behaviour' (e.g. when they crystallize / boil / dissolve) can be understood. My current research focuses on polydisperse systems that are out of equilibrium, to model the dynamics of mixing and demixing.
Micro-rheology

Rheology is usually conducted by applying large-scale shear to the fluid under investigation, by moving parallel plates. In micro-rheology, tiny sample volumes can be studied by analysing the motion of microscopic test-beads suspended in the fluid. This motion may be either Brownian, or driven by an applied magnetic field. In on-going work, in collaboration with Dr M Tassieri, we calculate ways to extract, from the correlated motions of many probe-particles, accurate measurements of the macroscopic visco-elastic shear moduli, which are unaffected by the finite-size boundary effects that usually plague such measurements. For details, click here.

Evolution

I investigate models of evolution, both numerically and analytically, specifically to answer the long-posed question of how altruism can ever arise by Darwinian evolution. This is a topic of current interest in biology, philosophy and games theory, and has recently become a hot topic in statistical mechanics. The standard methods of statistical mechanics are well suited to making progress in this area, where some over-simplified mathematical models have previously failed.
Driven Steady States
Currently, I am interested in the steady states that exist when complex fluids are subjected to a uniform shear flow, e.g. by two parallel plates that are moved relative to each other. Such steady states cannot be described by classical statistical thermodynamics, because they are not at equilibrium, but are "driven" by the work input at the boundaries. Some fluids "shear-thicken", i.e. become more viscous when they are sheared, while others shear-thin. Nematic liquid crystals (such as in LCDs) are composed of rod-like molecules that tend to allign with the direction of the flow. Some materials even separate into non-uniform bands when subjected to shear.

In the equilibrium case (without flux), theorists know how to construct informative simplified models of thermodynamic systems; they must respect the "principle of detailed balance" which ensures that the model system will operate in an unbiased manner that conforms to the known laws of statistical mechanics. For non-equilibrium systems such as complex fluids in shear flow, on the other hand, theorists have so far been guided only by intuition when constructing models, in the absence of any well developed non-equilibrium statistical mechanics.

But an exact theorem exists, relating to the statistics of motion in flowing systems. It governs the steady state motion of any sheared fluid that is stochastic (subject to thermal noise from its environment), ergodic (sufficiently mobile to thoroughly explore its space of available states) and microscopically reversible (contains normal particles with no sense of direction; this does not apply, for instance, to traffic flow).

The new theorem embodies a set of rules that put constraints on the modelling of non-equilibrium systems, ensuring self-consistency and no unwarranted bias. The resulting mathematical structure bears beautiful similarities to that of equilibrium statistical mechanics, and has as much potential for diverse and important applications. I believe that this development will shift the paradigm for theoretical modelling of non-equilibrium systems, removing the arbitrariness from the subject, and providing a foundation of rigorous statistical arguments. Potentially, the formalism should be able to explain much of the phase behaviour of fluids under flow.

This development has been timely, as there is rapidly increasing interest in exact non-equilibrium statistical mechanics, following a number of other recent exact theorems: "Fluctuation Theorems" have been discovered, quantifying the probability of momentary violations of the second law of thermodynamics, and "Non-equilibrium Work Theorems" have provided some exact criteria for the work done by non-equilibrium changes to the constraints on a thermodynamic system (with implications e.g. for experiments on forced unfolding of proteins). Also, the Lee-Yang theory of phase transitions has been generalised to a large class of non-equilibrium models. ( Richard Blythe is one of the authors of that work.) The subject area of non-equilibrium steady states now has huge potential for important discoveries.

My aim is now to explore the implications of the new theorems, by constructing simplified models, as well as making calculations for real systems (such as polymers under flow) that can be tested experimentally.



Click to hear an Isaac Newton Institute seminar on my research into non-equilibrium steady states of matter.

Selected Publications

(For a full list, click here.)


Numerical comparison of a constrained path ensemble and a driven quasisteady state
Milos Knezevic and R M L Evans,
Phys. Rev. E 89, 012132 (2014)
. Open access preprint: arXiv:1310.4384 [cond-mat.stat-mech]

Statistical mechanics far from equilibrium: Prediction and test for a sheared system
R M L Evans, R A Simha, A Baule and P D Olmsted,
Phys. Rev. E 81, 051109 (2010).

The effects of polydispersity and metastability on crystal growth kinetics
J J Williamson and R M L Evans,
Soft Matter, Advance Article (2013).
Open access preprint at arxiv.org/abs/1208.3804

Direct conversion of rheological compliance measurements into storage and loss moduli
R M L Evans, Manlio Tassieri, Dietmar Auhl, and Thomas A. Waigh,
Phys. Rev. E 80, 012501 (2009).

Invariant quantities in shear flow
A Baule and R M L Evans,
Phys. Rev. Lett. 101, 240601 (2008).

Dynamics of Semiflexible Polymer Solutions in the Highly Entangled Regime
Manlio Tassieri, R. M. L. Evans, Lucian Barbu-Tudoran, G. Nasir Khaname, John Trinick, and Tom A. Waigh,
Phys. Rev. Lett. 101, 198301 (2008).

Rules for transition rates in nonequilibrium steady states ,
R M L Evans,
Phys. Rev. Lett. 92, 150601 (2004).

Universal law of fractionation for slightly polydisperse systems
R M L Evans, D J Fairhurst and W C K Poon,
Phys. Rev. Lett. 81, 1326-1329 (1998).

Colloid-polymer mixtures at triple coexistence: Kinetic maps from free-energy landscapes,
W C K Poon, F Renth, R M L Evans, D J Fairhurst, M E Cates and P N Pusey,
Phys. Rev. Lett. 83, 1239-1242 (1999).

Correlation length by measuring empty space in simulated aggregates,
R M L Evans and M D Haw,
Europhys. Lett. 60(3), 404-410 (2002).

Role of metastable states in phase ordering dynamics,
R M L Evans, W C K Poon and M E Cates,
Europhys. Lett. 38, 595-600 (1997).

Detailed balance has a counterpart in non-equilibrium steady states,
R M L Evans,
J. Phys. A: Math. Gen. 38, 293-313 (2005).


My PhD thesis:

A theoretical study of orientational order in liquid crystal vesicles,
R. M. L. Evans,
University of Manchester Thesis (1995).
chapter 1 chapter 2 chapter 3 chapter 4 appendices

For a full list of publications, click here.




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