Overview 1. Introduction to statistical mechanics: Partition - PowerPoint PPT Presentation

Phase transitions for particles in R 3 Sabine Jansen LMU Munich Konstanz, 29 May 2018

Overview 1. 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

Mechanics – Thermodynamics – Statistical Mechanics ◮ Classical mechanics: ODEs for particle positions and velocities x 1 , . . . , x N , v 1 = ˙ x 1 , . . . , v N = ˙ x N . E.g.: � � � ¨ x i ( t ) = − ∇ V x i ( t ) − x j ( t ) i = 1 , . . . , 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!) � N Temperature T ∝ 1 1 � � x 2 2 ˙ average kinetic energy. i N i =1 Particle number N of the order of 10 23 : large.

Averages I: microcanonical ensemble ◮ Average over long periods of time: � ∆ t � N 1 1 � � x 2 2 ˙ i ( s ) d s , ∆ t → ∞ . ∆ t 0 i =1 Ergodic theory? Time average = space average ? ... ◮ Average over invariant probability distribution for positions and velocities: N particles in box Λ = [0 , L ] 3 . Energy N 1 ( x , v ) ∈ Λ N × ( R 3 ) N . � 2 v 2 � H ( x , v ) := i + v ( x i − x j ) , i =1 1 ≤ i < j ≤ N Uniform distribution on energy shell E − δ E ≤ H ≤ E � N 1 � � 2 v 2 i E , N , Λ; δ E i =1 � N Ω( E , N , Λ; δ E ) × 1 1 � 1 � � 2 v 2 := 1 { E − δ E ≤ H ( x , v ) ≤ E } d x d v . i N ! Λ × R 3 N i =1 Ω( E , N , Λ; δ E ) = Normalization constant = partition function.

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; δ E ) := 1 � ( x , v ) ∈ Λ N × R 3 N � �� � � E − δ E ≤ 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) = 1 � � Λ N × R 3 N exp − β H ( x , v ) d x d v . N ! β > 0 inverse temperature. Energy exchanged with heat bath... ◮ Grand-canonical: Energy E random, particle number N random. Part. fct. ∞ exp( βµ N ) � � � � Ξ( β, µ, Λ) = 1 + Λ N × R 3 N exp − β H ( x , v ) d x d v . N ! N =1 µ ∈ R chemical potential. Heat bath + particle reservoir.

Thermodynamic limit. Entropy and free energy Particle number N ≈ 10 23 very large ⇒ approximate with limit N → ∞ . Keep parameters β, µ and particle and energy densities fixed. | Λ | = L 3 → ∞ , N → ∞ , | 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 1 | Λ | log Ω( E , N , Λ; δ E ) entropy 1 f ( β, ρ ) = − lim β | Λ | log Z ( β, N , Λ) free energy 1 p ( β, µ ) = lim β | Λ | log Ξ( β, µ, Λ) pressure . Limits exist under suitable assmptions on vV ( x i − x j ). Change of ensembles with Legendre transforms: u − β − 1 s ( u , ρ ) � � � � f ( β, ρ ) = inf , p ( β, µ ) = sup µρ − f ( β, ρ ) . u ∈ R ρ> 0 Convexity ⇒ inverse relations as well.

Derivatives vs. expected values Observation: � H exp( − β H ) d x d v − 1 ∂β log Z ( β, N , Λ) = 1 ∂ exp( − β H ) d x d v . � | Λ | | Λ | ⇒ if limits and differentiation can be exchanged: � H ( x , v ) ∂ � � � β f ( β, ρ ) = lim | Λ | ∂β 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 = � 1 2 v 2 i + U ( x 1 , . . . , x N ) ⇒ Integrals factorize, v 1 , . . . , v N normally distributed, � N 1 β, N , Λ = 3 2 N β − 1 = 3 � � 2 v 2 2 NT i i =1 β − 1 ∝ average kinetic energy → temperature.

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.

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 100 ◦ 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 ’71 ◮ Four-body interaction + pair potential, van der Waals theory Lebowitz, Mazel, Presutti ’99

Low temperature and density: conjectures Conjecture I : ∃ ρ 0 > 0 und curve ρ sat ( β ) with ρ sat ( β ) ≈ exp( − const β ) → 0 at β → ∞ ( T → 0) such that ρ �→ f ( β, ρ ) ◮ 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 ∂µ ( β, µ ) has jump discontinuity at µ sat ( β ). Phase transition at µ = µ sat ( β ). Pressure as function of µ has a kink.

3 Low temperature and low density: a partial result Under suitable assumptions on pair potential V ( x − y ): 1 � e ∞ := lim inf V ( x i − x j ) ∈ ( −∞ , 0) . N N →∞ x 1 ,..., x N ∈ R 3 1 ≤ i < j ≤ N Theorem (J ’12) ∃ ν ∗ > 0 such that as β → ∞ ( µ fixed): ∂ p ∂µ ( β, µ ) = O (exp( − βν ∗ )) → 0 ∀ µ < e ∞ : lim inf ∂ p ∀ µ > e ∞ : ∂µ ( β, µ ) ≥ ρ 0 > 0 . Step towards kink of µ �→ p ( β, µ ) at µ ≈ e ∞ . Theorem (J ’12) Suppose there is a phase transition at ρ sat ( β ) → 0 . Then µ sat ( β ) = e ∞ + O ( β − 1 log β ) ( β → ∞ ) . Tells us where to look for phase transitions.

4 Proof ingredient I: cluster expansions Set z = exp( βµ ). Remember: � ∞ � z N 1 � � p ( β, µ ) = lim β | Λ | log 1 + Λ N × R 3 N exp( − β H ) d x d v . N ! | Λ |→∞ N =1 Right-hand side is power series in z : ∞ � b n , Λ ( β ) z n , p ( β, µ ) = lim z = exp( βµ ) . | Λ |→∞ n =1 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’ 12 1 lim inf β log R ( β ) = e ∞ . β →∞

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 | k ρ k ≤ ρ . k =1 f ( β, ρ, ( ρ k )) = restricted free energy � β | Λ | log 1 1 ∀ k : N k ( x ) � � e − β H ( x , v ) d x d v � � f β, ρ, ( ρ k ) = − lim ≈ ρ k 1 N ! | Λ | N k ( x 1 , . . . , x N ) = number of droplets with k particles. 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., K¨ onig, Metzger ’11; J., K¨ onig ’12

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.