Phase transitions for particles in R 3

Transcription

1 Phase transitions for particles in R 3 Sabine Jansen LMU Munich Konstanz, 29 May 208

2 Overview. Introduction to statistical mechanics: Partition functions and statistical ensembles 2. Phase transitions: conjectures 3. Mathematical results: a step in the right direction 4. Proof ideas

3 Mechanics Thermodynamics Statistical Mechanics Classical mechanics: ODEs for particle positions and velocities x,..., x N, v = ẋ,..., v N = ẋ N. E.g.: ẍ i (t) = ( ) V x i (t) x j (t), i =,..., N. j i N interacting particles, pair potential v(x y). Thermodynamics: No modelization of individual particles. Instead, macroscopic quantities like pressure p, temperature T, energy, heat, entropy... Statistical mechanics: interpret macroscopic quantities of thermodynamics as averages of microscopic quantities from mechanics. E.g. absolute temperature (Kelvin, not Celsius!) Temperature T N N i= Particle number N of the order of 0 23 : large. 2 ẋ i 2 average kinetic energy.

4 Averages I: microcanonical ensemble Average over long periods of time: t t 0 ( N i= ) 2 ẋ i 2 (s) ds, t. Ergodic theory? Time average = space average?... Average over invariant probability distribution for positions and velocities: N particles in box Λ = [0, L] 3. Energy H(x, v) := N i= 2 v 2 i + i<j N v(x i x j ), (x, v) Λ N (R 3 ) N. Uniform distribution on energy shell E δe H E N i= := 2 v i 2 E,N,Λ;δE Ω(E, N, Λ; δe) ( N ) N! Λ R 3N 2 v i 2 {E δe H(x,v) E} dxdv. i= Ω(E, N, Λ; δe) = Normalization constant = partition function.

5 Averages II: canonical and grand-canonical ensemble Box Λ = [0, L] 3 R 3. Microcanonical: Energy E fixed, particles number N fixed. Partition function Ω(E, N, Λ; δe) := { } (x, v) Λ N R 3N. E δe H(x, v) E N! Isolated system no energy or particle exchanged with environment. Other invariant measures (ensembles): Canonical: Energy E random, particle number N fixed. Partition function Z(β, N, Λ) = ( ) exp βh(x, v) dx dv. N! Λ N R 3N β > 0 inverse temperature. Energy exchanged with heat bath... Grand-canonical: Energy E random, particle number N random. Part. fct. Ξ(β, µ, Λ) = + N= exp(βµn) N! ( ) exp βh(x, v) dx dv. Λ N R 3N µ R chemical potential. Heat bath + particle reservoir.

6 Thermodynamic limit. Entropy and free energy Particle number N 0 23 very large approximate with limit N. Keep parameters β, µ and particle and energy densities fixed. N, Λ = L 3, E, β, µ, ρ = N Λ, u = E Λ fest. Asymptotics of partition functions define physically relevant quantities. Boltzmann entropy S = k log W. s(u, ρ) = lim log Ω(E, N, Λ; δe) Λ entropy f (β, ρ) = lim log Z(β, N, Λ) β Λ p(β, µ) = lim log Ξ(β, µ, Λ) β Λ pressure. free energy Limits exist under suitable assmptions on vv (x i x j ). Change of ensembles with Legendre transforms: ( f (β, ρ) = inf u β s(u, ρ) ) ( ), p(β, µ) = sup µρ f (β, ρ). u R Convexity inverse relations as well. ρ>0

7 Derivatives vs. expected values Observation: Λ β H exp( βh)dxdv log Z(β, N, Λ) =. Λ exp( βh)dxdv if limits and differentiation can be exchanged: ( ) H(x, v) βf (β, ρ) = lim β Λ β,n,λ β-derivative average energy density. Similarly N µ p(β, µ) = lim Λ β,µ,λ µ-derivative average particle density. Kinetic energy in the canonical ensemble: H = v 2 2 i + U(x,..., x N ) Integrals factorize, v,..., v N normally distributed, N i= 2 v i 2 = 3 β,n,λ 2 Nβ = 3 2 NT β average kinetic energy temperature.

8 Formalism statistical mechanics (classical, equilibrium): Description of finite systems by probability measures on phase space (positions + velocities). Different measures (ensembles) possible. E.g. uniform distribution on energy shell. Normalization constants (partition functions) are physically relevant. E.g. S = k log W. Micro-macro: For a given pair potential V (x i x j ), the entropy, free energy and pressure are uniquely defined by asymptotics of high-dimensional integrals. microscopic definition of macroscopic thermodynamic potentials.

9 2 Phase transitions Question: are the entropy, free energy and pressure analytic functions? Kinks? strictly convex (concave) or with affine pieces? Interpretation: Non-analyticity = Phase transition. From ice to liquid water to vapor. Small increase of temperature near boiling point 00 C abrupt change of material properties. Can only happen in the limit N, Λ! Math: open. Existence of phase transitions proven for Particle on lattices / Ising model on Z 2 Peierls 36 Widom-Rowlinson model: multi-body interaction Ruelle 7 Four-body interaction + pair potential, van der Waals theory Lebowitz, Mazel, Presutti 99

10 Low temperature and density: conjectures Conjecture I: ρ 0 > 0 und curve ρ sat(β) with such that ρ f (β, ρ) ρ sat(β) exp( constβ) 0 at β (T 0) analytic and strictly convex in ρ < ρ sat(β), affine in ρ sat(β) ρ ρ 0. Phase transition at ρ = ρ sat(β). Free energy as a function of density ρ has affine piece. Conjecture II: µ 0 > 0 and curve µ sat(β) such that µ p(β, µ) analytic in µ < µ sat(β) analytic in µ sat(β) µ µ 0 µ p (β, µ) has jump discontinuity at µsat(β). µ Phase transition at µ = µ sat(β). Pressure as function of µ has a kink.

11 3 Low temperature and low density: a partial result Under suitable assumptions on pair potential V (x y): Theorem (J 2) e := lim N N inf x,...,x N R 3 i<j N ν > 0 such that as β (µ fixed): µ < e : V (x i x j ) (, 0). p µ (β, µ) = O(exp( βν )) 0 µ > e : lim inf p (β, µ) ρ0 > 0. µ Step towards kink of µ p(β, µ) at µ e. Theorem (J 2) Suppose there is a phase transition at ρ sat(β) 0. Then µ sat(β) = e + O(β log β) (β ). Tells us where to look for phase transitions.

12 4 Proof ingredient I: cluster expansions Set z = exp(βµ). Remember: ( p(β, µ) = lim Λ β Λ log z N ) + exp( βh)dx dv. N! Λ N R 3N Right-hand side is power series in z: N= p(β, µ) = lim Λ n= b n,λ (β)z n, z = exp(βµ). Mayer expansion, cluster expansion. Known: For every fixed box, radius of convergence R Λ (β) > 0. Bounds that are uniform in Λ R(β) = lim inf Λ R Λ(β) > 0. Pressure is analytic in z = exp(βµ) < R(β). Asymptotics of coefficients and radius of convergence J 2 lim inf β log R(β) = e. β

13 Proof ingredient II: droplet sizes Assume: pair potential v has compact support. Variational representation f (β, ρ) = inf {f ( ) } β, ρ, (ρ k ) k N kρ k ρ. f (β, ρ, (ρ k )) = restricted free energy f ( β, ρ, (ρ k ) ) = lim β Λ log N! ( k : k= N k (x) Λ N k (x,..., x N ) = number of droplets with k particles. ρ k )e βh(x,v) dx dv Sator, Phys. Rep. 376 (2003) At low temperature and low density: have good bounds for restricted free energy, deduce bounds for free energy f (β, ρ). J., König, Metzger ; J., König 2

14 Summary & Outlook Summary: Asymptotics of high-dimensional, parameter-dependent integrals functions of parameters. Analytic? Question still open, but partial mathematical results consistent with physics. Connections: Probability theory: large deviations, Gibbs measures, point processes, percolation... Analysis: energy minimizers and energy landscape, crystallization, Wulff shapes. Combinatorics: cluster expansions related to counting connected graphs; random combinatorial structures. Also: Dynamics of phase transitions: nucleation barriers? Metastability? e.g. for Markov processes with prescribed invariant measure.