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Applications of non-equilibrium statistical mechanics

Applications of non-equilibrium statistical mechanics

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Applications of non-equilibrium statistical mechanics

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  1. Applications of non-equilibrium statistical mechanics Differences and similarities with equilibrium theories and applications to problems in several areas. BirgerBergersen Dept. Of Physics and Astronomy University of British Columbia

  2. Acknowledgements These slides are based on lectures held at Beijing Normal University May 2012. I am grateful to my hosts at the university, and in particular to Professors Jinshan Wu and Zengru Di for making my stay in Beijing thoroughly enjoyable. Some additions have been made to the original material and I plan to add more material.

  3. Overview • Properties at equilibrium • Steady states • Beyond the steady state • Stochastic processes and Brownian motion • Non-equilibrium distributions • Correlated random walks • Continuous time walks

  4. Properties at equilibrium • Macro & micro-states, micro-canonical & canonical ensembles • Variational principles • Thermodynamic derivatives • State and process variables, the problem with utility • Extensive and intensive variables, Gibbs Duhem relation • Gibbs and grand canonical ensembles • Equilibrium fluctuations • Detailed balance • Equations of state • Phase transitions, van der Waals fluids • Symmetry breaking • Information theory and entropy

  5. Insulating walls Starting point for equilibrium statistical physics: Macro-state average over allowed micro-states. Micro-canonical ensemble: Canonical ensemble: Variational principle: entropy Smaximumsubject to constraints N,E,V fixed Temperature T, pressure P, chemical potential μfluctuating. N,E,V specified Helmholtz free energy A=E-TSminimum,subject to T,V,N fixed. Competition between energy and entropy. Entropy S , energy E, pressure P, chemical potential μ fluctuating. T,V,N specified Heat bath at temperature T

  6. Thermodynamic derivatives of one component system: In the micro-canonical ensemble: Energy of a state is a function of the S,V,N with an exact differential The identification can be thought of as definitionsofT,P,μ . Exact differential implies Maxwell identities such as The first law of thermodynamics states that where Q stands for heat added to system and W work done by system. Q and W are process variables not state variables. Knowledge of the initial and final states of a process does not allow a determination of the heat and work of the process.

  7. The distinction is between state and process variables is important for utility theory in economics and game theory. In neoclassical economics and conventional game theory the preferences are fixed and utility can be treated as a state variable, to be maximized much like a free energy is minimized in equilibrium theory. But this view has always been controversial. For a historical perspective see Mirowski[3]. On the other hand in the prospect theory of Tversky and Kahneman[4] utility is highly dependent on framing which is in turn depends on context and history. Utility must then be treated as a process variable. Utility theory assigns values to outcomes, but if our goals are to obtain status and influence, process may be more important than outcome. Also many of our tastes and preferences are learned and history dependent. For my personal view on the subject see [5].

  8. Gibbs-Duhemrelation Equilibrium thermodynamic variables such as S,V,N,E are extensive i.e. they are proportional to system size. Other variables such as T,P,μ are intensive and independent of size if system large. Let us rescale a one component system by a factor of λ E(λS, λV, λN)= λ E(S,V,N) Differentiating both sides with respect to λ gives TS-PV+μN=E which is the Gibbs Duhem relation. The environment surrounding the atoms and molecules in ordinary matter is local and does not depend on size of object. This sometimes referred to as the law of existence of matter [6]. Evolving systems such as ecologies, organisms, the internet, earthquake fault systems, do not follow this law, and they are not equilibrium systems, nor are astronomical objects held together by gravity extensive or in equilibrium.

  9. P • Gibbs ensemble G=E-TS+PV = μN minimum V, μ, S fluctuating • Grand canonical ensemble Mathematically the different free energies are related by Legendre transforms. Heat bath P,T,N specified Porous walls N,P,S fluctuating Grand potential Ω=-PV minimum μ, V, T specified Heat bath

  10. Canonical ensemble dA =-SdT -PdV +μdN Example isothermal atmosphere: If atmospheric pressure at height h=0 is p(0) and m the mass of a gas molecule But, real atmospheres are not isothermal and not in equilibrium.

  11. Canonical fluctuations: • ...

  12. I assume that what we done so far are things you have seen before, but to refresh your memory let us do some simple problems: Problem 1: In the micro-canonical ensemble the change in energy E, when there is an infinitesimal change in the control parameters S,V, N is dE= TdS-PdV+μdN So that T=∂E/ ∂S, P=- ∂E/ ∂V, μ= ∂E/ ∂N a. Show that if two systems are put in contact by a conducting wall their temperatures at equilibrium will be equal. b. Show that, if the two systems are separated by a frictionless piston, their pressures will be equal at equilibrium. c. Show that if they are separated by a porous wall the chemical potential will be equal on the two sides at equilibrium.

  13. Solution: Instead of considering the state variable E as E(S,V,N) we can treat the entropy as S(E,V,N) dS=E/T dE +P/T dV –μ/ T dN Or 1/T=∂S/ ∂E Let E=E₁+E₂ be the energy of the compound system and the entropy is S=S ₁+S₂. At equilibrium the entropy is maximum or Since ∂E₁ / ∂E₂=-1 we find 1/T₁ =1/T₂ or T₁ =T₂ . Similar arguments for the case of the piston tells us that at equilibrium P₁/T₁ = P₂/T₂. But, since T₁ =T₂, P₁= P₂. The same argument tells us that μ₁= μ₂. Note that if the two systems are kept at different temperatures, the entropy is not maximum when the pressures are equal, as required by mechanical equilibrium!

  14. Problem 2: If q and p are canonical coordinated and momenta the number of quantum states between q and q+dq, p and p+dp are dpdq/h, where h is Planck’s constant. If a particle om mass m is in a 3D box of volume V its canonical partition function is The partition function for an ideal gas of N identical particles is N! represents the # of permutations of states and particle labels. If there are more particles than available states the number of combinations should be used instead (see problem 2.13 of [1]). a. Find expressions for <A>, <S> , < E> and <P>. b. Two ideal gases with N₁=N₂=N molecules occupy separate volumes V₁=V₂=V at the same temperature. They are then mixed in a volume 2V. Find the entropy of mixing. Show that there is no entropy change if the molecules are of the same kind!

  15. Solution: We have For large N we can use Stirlings formula ln N!=N ln N-N and find The entropy can be obtained from S=-∂A/ ∂T, and Similarly E=A+TS=3nkT/2 and P=-∂A/ ∂V=NkT If the two gases are different the change in entropy comes about because V→2V for both gases. Hence, ∆S=2Nk ln 2 If the two gases are the same there is no change in the total entropy.

  16. Problem 3. Maxwell speed distribution • Assuming that the number of states with velocity components between • is proportional to and that the energy of a molecule with speed is find the • probability distribution for the speed. • Find the average speed • The mean kinetic energy • The most probable speed i.e. the speed for which

  17. Solution: Going to polar coordinates we find that To find the proportionality constant we use after some algebra: Integrating we find

  18. Principle of detailed balance: The canonical ensemble is based on the assumption that the accessible states j have a stationary probability distribution π(j)=exp(-βE(j))/Z(β) This distribution is maintained in a fluctuating environment in which the system undergoes a series of small changes with probabilities P(m←j). For the distribution to be stationary it must not change by transitions in and out of state π(m)=∑ P(j←m) π(m)= ∑ P(m←j) π(j) This is satisfied by requiring detailed balance: P(j ←m)/P(m ←j)= π(j)/ π(m)=exp(-β(E(j)-E(m)) In an equilibrium simulation we get the correct equilibrium properties, by satisfying detailed balance and making sure there is a path between all allowed states (Monte Carlo method).. We can ignore the actual dynamics. Very convenient! j j

  19. Equations of state: The equilibrium state of ordinary matter can be specified by a small number of control variables. For a one component system the Helmholtz free energy is A=A(N,V,T). Other variables such as the pressure P, the chemical potential μ, the entropy S are given by equations of state μ=∂A/ ∂N, P= - ∂A/ ∂V, S=- ∂A/ ∂T An approximate equation of state describing non-ideal gases Is the van der Waals equation which obtains from the free energy Comparing with the monatomic ideal gas expression we interpret the constant a as a measure of the attraction between molecules, outside a hard core volume b

  20. Equilibrium phase transitions For special values of the control parameters(P,N,T in the Gibbs ensemble) the order parameters (V,μ,S) changes abruptly. P and T for a single component system when solids/liquid and gas/liquid co-exists at equilibrium is shown at the top while the specific volume v=V/N for a given P is below. The gas/liquid and solid/liquid transition are 1st order (or dis- continuous). The critical endpoint of the latter is a 2nd order (or continuous ). Specific heat and fluctuations diverge at critical point.

  21. Combining the P-V and P-T plots into a single 3D plot.

  22. Problem 4: Van der Waals equation. • Express the equation as a cubic equation in v=V/N. • Find the values for which the roots coincide. • Rewrite the equation in terms of the reduced variables. • the resulting equation is called the law of corresponding states. • d. Discuss the behaviour of the isotherms of the law of corresponding states.

  23. Solution: • Multiplying out and dividing by a common factor we find • When the roots are equal this expression must be on the form • comparing term by term we find after some algebra • Again, after some algebra we find in terms of the new variables the law of corresponding states • There are no free parameters. All van der Waals fluids are the same! • We next plot p(v,t) for different values of t

  24. For t<1 there is a range of pressures p₁<p<p₂ with 3 solutions . The points p₁ and p₂ are spinodals. In the region between them the compressibility -∂P/∂V is negative and the system is unstable. The equilibrium state has the lowest chemical potential μ ←t=1.1 P p₂ t=1 → ←t=0.9 p₁ We have μ=(A+PV)/N so that dμ=(SdT+VdP)/N μ₂- μ₁=∫²₁=∫VdP/N The equilibrium vapor pressure is when the areas between the horizontal line and the isotherms are equal. Between the equilibrium pressure and the spinodal there is a metastable region. Dust can nucleate rain drops. If particles too small surface tension prevents nucleation. VdW theory qualitatively but not quantitatively ok.

  25. Symmetry breaking: Alben model. A large class of equilibrium systems characterized by a high temperature symmetric phase and low temperature broken symmetry phase. Some examples: Magnetic models: Selection of spin orientation of “spins”, Ising model, Heisenberg model, Potts model.... Miscibility models: High temperature solution phase separates at lower temperatures. Many other models in cosmology, particle and condensed matter. The Alben model [7]is a simple mechanical example. An airtight piston of mass M separates two regions, each with N ideal gas molecules at temp. T The piston is in a semicircular tube of cross section a. The order parameter of the is the angle Φ of the piston.

  26. The free energy of an ideal gas is A=-NkT ln (V/N)+terms independent of V, N yielding the equation of state PV=-∂A/ ∂V=NkT We add the gravitational potential energy to obtain A=M gR cos φ- NkT(ln [aR(π/2+ φ)/N]+ln [aR(π/2+-φ)/N] Minimizing with respect to φ gives 0= ∂A/ ∂ φ=-MgR sin φ-8NkT φ/(π²-4 φ²) This equation has a nonzero solution corresponding to minimum for T<T₀=MgR π²/(8Nk). The resulting temperature dependence of the order parameter is shown below to the left. To the right we show the behaviour if the amount of gas on the two sides is N(1±0.05)

  27. At equilibrium: • Second law: Variational principles apply: entropy maximum subject to constraints → free energy minimum . System stable with reversible, Gaussian fluctuations (except at phase transitions, continuous or discontinuous) → No arrow of time, detailed balance. • Zeroth law: Pressure, temperature chemical potential constant throughout system. • No currents (of heat, charge, particles) • Thermodynamic variables: Extensive (proportional to system size, N,V,E,S), or intensive (independent of size, μ,T,P). Law of existence of matter. • Typical fluctuations proportional to square root of size.

  28. Information theory approach [1][2]

  29. Steady states • Steady states vs. equilibrium • Equilibrium “economics”, wealth distribution, Pareto tail • Failure of variational principle • Linear response: Ohm’s, Fick’s and Stokes laws. • Thermoelectric, Soret and thermomolecular effects • Onsager relations • Approach to equilibrium, le Chatelier’s principle

  30. Steady state vs. equilibrium: • Steady states are time independent, but not in equilibrium. Heat current Electric current T+Δ T J Q No zeroth law! Gradients of temperature, chemical potential and pressure allowed. Osmosis Salt water Fresh water

  31. Energy Raw materials Labour Capital Economy Economy Waste Products Wages Profits If no throughput there is no economy! Economic “equilibrium” an oxymoron. Not everyone agrees [9] !

  32. Equilibrium wealth distribution: N = # of citizen >>1, w = W/N= average wealth x= wealth of individual Most probable wealth distribution [9]: Equilibrium approached, if funds are spent and earned in many small random amounts. We can imagine this taking place if wealth changes by the buying and selling of goods and services . Average wealth analogous to thermodynamic temperature. Problem: Wealth inequality varies from society to society, and over time! Wealth of the very rich follows power law [10]. Most people’s earnings come from salaries, which tend to change by multiplicative process. Wealth generated by production of goods a non-equilibrium process. Income distribution similar but inequality tends to be less pronounced.

  33. The power law part of the income or wealth distribution is commonly referred to as the Pareto tail after Vilfredo Pareto who wrote in 1887: • In all places and at all times the distribution of income in a stable economy, when the origin of measurement is at a sufficiently high income level, will be given approximately by the empirical formula • where y is the number of people having an income x or greater and v is approximately 1.5. • To my knowledge this is the first claim in the literature of universality of critical exponents! Empirically the exponent is probably not universal. There is also some indication that the exponent is different for the very and the extremely rich. Note that the number of people in the tail will not be universal, so the inpact on overall inequality will vary from society to society.

  34. In non-equilibrium steady state entropy not maximized (Subject to constraint). Example: Equal gas containers are connected by a capillary. One is kept at a higher temperature than the other. If the capillary not too thin system will reach mechanical equilibrium (constant pressure). The cold container will then have more molecules than the hot one. The entropy per molecule is higher in hot gas P, V Tc P, V Th If entropy maximum subject to temperature constraint molecules would move from cold to hot. This does not happen! Chemical potential will be different on the two sides in steady state. Steady states not governed by variational principle! Zeroth law does not apply!

  35. Linear response • Ohm’s law: Electrical currentV=R IV=voltage, R= resistance, I=current • Fick’s law of diffusion • Stokes law for settling velocity Many similar laws!

  36. More subtle effects [11]: Temperature gradient in bimetallic strip produces electromotive force → Seebeck effect Electric current through bimetallic strip leads to heating at one end And cooling at the other → Peltier effect Dust particle moves away from hot surface towards cooler region. →Thermophoresis.Particles in mixture may selectively move towards hotter or cooler regions → Ludwig-Soreteffect: in rarefied gases a temperature gradient may a pressure gradient → thermo-molecular effect

  37. Le Chatelier’s principle Any change in concentrations, volume or pressure is counteracted by shift in equilibrium that opposes change. If a system is disturbed from equilibrium , it will return monotonically towards equilibrium as disturbance is removed (no overshoot). This is also suggested by the fact that the Onsager coefficients have real eigenvalues. In neoclassical economics it has been suggested that if supply exceeds demand, prices will fall, production cut and equilibrium restored. A similar process should occur if demand exceeds supply. But is this true?

  38. Beyond the steady state • Belousov –Zhabotinsky reaction • Rate equation approach • The brusselerator • SIR model of epidemiology • Lotka-Volterra type models, paradox of enrichment • Logistic models, competitive exclusion and paradox of plankton. • Delay effects. Hutchinson’s equation & modifications. • Inertia:Transition to turbulence, dimensional analysis. Moody diagrams. • Traffic flow problems. Braess paradox.

  39. Belousov-Zhabotinsky reaction In the 1950s Boris Pavlovich Belousov discovered an oscillatory chemical reaction in which Cerium IV ions in solution is reduced by one chemical to Ce III and subsequently oxidized to Ce IV causing a periodic color change. Belousov was unable to publish his results, but years later Anatol Zhabotinsky repeated the experiments and produced a theory . By now many other chemical oscillators have been found.

  40. Rate equation approach: Chemistry’s law of mass action allows one to find steady states (equivalent to equations of state), and the approach to equilibrium. Consider the reaction X+Y↔Z in which 2 atoms combine to form a molecule Z. Concentrations:x,y,z. Rate constants: k₁ (→), k₂ (←). Rate equations: The steady state is obtained by putting the time derivatives to zero The approach to steady state can then be obtained by solving the differential equations.

  41. The brusselator: Ilya Prigogine and co-workers developed a simple model which shows similar behavior to the BZ system. Usually it is studied as a spatial model in which the reactant diffuse in addition to reacting. We will simplify to the well stirred case in wich concentrations are uniform [12] . The model consists of the fictional set of reactions. A→ X 2X+Y→ 3X B+X→ Y+D X→ E The concentrations a,b of A,Bare kept constant, D and E are eliminated waste products. The reactions are irreversible and for simplicity the rate constants are set to 1. The corresponding rate equations are dx/dt=a+x²y-bx-x; dy/dt=bx-x²y

  42. The model has a fixed point (steady state) obtained by setting reaction rates to zero 0=a+x²y-bx-x; 0=bx-x²y; We find x₀=a, y₀=b/a In order to study the stability properties we linearize about the fixed point. x=a+ξ; y=b/a +η; We find d ξ/dt=(b-1) ξ+a² η+...; dη/dt=-bξ-a²η+...; The solution is a linear combination of exponentials with exponents eigenvalues of We find that eigenvalues are complex (oscillatory solution) if 1+a²+2a>b>1+a²-2a They have a negative real part (stable fixed point) if b<1+a²

  43. A linear stability analysis of the fixed point in parameter space is shown in the picture to the right. Trajectories of the concentrations x and y are shown below. ↘ ↘ ↘ ↘ Le Chatelier’s principle does not apply because the system is driven, A,B must be kept constant. Reactions are irreversible.

  44. SIR-model of epidemiology: Divide population into susceptible, S, infected, I, removed (immune), R. Put Ω=S+I+R. The following processes take place: Susceptibles are born or introduced at rate Ωγ(1-ρ) Infecteds are introduced at rate Ωγρ Susceptibles, become infected at rate βSI/Ω Susceptibles die at the rate γS Infecteds are removed at rate λI, by death or immunity Assume that total population Ωis kept constant. Introduce concentrations ф=S/Ω, ψ=I/ Ω. Obtain rate equations: dф/dt= γ(1- ρ)-βфψ- γ ф dψ/dt= γρ+ βфψ-λψ Conditions for steady state obtained by putting time derivatives to zero. Make one more simplification and put ρ=0

  45. Condition for steady state: • 0=γ-βфψ- γ ф • 0= βфψ-λψ • There are 2 solutions • ψ=0, ф=1 • Ψ=λ/β, ф= γ/(λ+γ) • I leave it as an exercise to show that (i) is stable when c= β/ γ<1 while (ii) is stable when c>1. In that case the system will approach the steady state through damped oscillations. It is easy to extend model to several sub-populations, or diseases. Fluctuations can be also be incorporated.

  46. Problem 5: Lotka-Volterramodel • Big fish eat little fishes to survive. The little fish reproduce at a certain net rate, and die by predation. The rate model rate eqs are • Simplify the model by introducing new variables • b. Find the steady states of the populations and their stability. Discuss the solutions to the rate equations. • c. In the model, the populations of little fish grows without limits if the big fish disappear. Modify the little fish equation to add a logistic term, with K the carrying capacity • How does this change things? • d. If the population of big fish becomes too small it may become extinct. Modify model to include small immigration rates. Neglect the effect of finite carrying capacity.

  47. Solution: a. With the substitutions the equations become dy/dτ=-ry(1-x); dx/dτ=x(1-y) b. There are two steady states {x=0,y=0}, and {x=1,y=1}. The former is a saddle point (exponential growth in the x-direction, and exponential decay of the big fishes). Linearizing around the second point x=1+ ξ, y=1+ηyields dη/dτ=r ξ, dξ/dτ=-η which are the equations for harmonic motion around a center. The full differential equations can be solved exactly. Dividing the two equations and cross multiplying This equation can easily be integrated. r=1 ↓ ↓