Phase Transitions for Dilute Particle Systems with Lennard-Jones Potential

Transcription

1 Weierstrass Institute for Applied Analysis and Stochastics Phase Transitions for Dilute Particle Systems with Lennard-Jones Potential Wolfgang König TU Berlin and WIAS joint with Andrea Collevecchio (Venice), Peter Mörters (Bath) and Nadia Sidorova (London) supported by the DFG-Forschergruppe 78 Analysis and Stochastics in Complex Physical Systems Mohrenstrasse Berlin Germany Tel Rome, University Tor Vergata, 7 November 200

2 Background Objective: study the transition between gaseous and solid phase in the thermodynamic limit for interacting many-particle systems. Very difficult at positive temperature and positive particle density. We study a dilute system at vanishing temperature. Extreme temperature choices: fixed positive temperature (inter-particle distance diverges, no interaction felt) zero temperature (rigid macroscopic structure emerges). We study the transition between these two situations. Questions: What is the critical scale? What is the emerging microscopical structure? What is the emerging macroscopical structure? We consider a non-collapsing dilute classical particle system with stable interaction. Dilute Particle Systems with Lennard-Jones potential Rome, University Tor Vergata, 7 November 200 Page 2 (4)

3 Energy Energy of N particles in R d : N V N (x,...,x N ) = v ( x i x j ), for x,...,x N R d. i, j= i j Pair-interaction function v: [0, ) (, ] of Lennard-Jones type: Lennard-Jones potential v(r) = r 2 r 6 examples of our potentials short-distance repulsion (possibly hard-core), preference of a certain positive distance, bounded interaction length. Dilute Particle Systems with Lennard-Jones potential Rome, University Tor Vergata, 7 November 200 Page 3 (4)

4 Dilute System N particles in the centred box Λ N R d with Volume Λ N N, i.e., vanishing particle density ρ N = N/ Λ N. Partition function with inverse temperature β (0, ), Z N (β,ρ N ) := { } dx...dx N exp βv N (x,...,x N ). N! Λ N N Idea: Couple inverse temperature β = β N with particle density ρ N 0 such that log = c (0, ) β N ρ N is constant. Then energic and entropic forces compete on the same, critical scale, and determine the behaviour of the system. (Example: β N logn and Λ N = N α with α >.) Free energy per particle: Ξ(c) = lim logz N (β N,ρ N ). N Nβ N Large c = entropy wins, i.e., typical inter-particle distance diverges, Small c = interaction wins, i.e., crystalline structure in the particles emerges. How does the crystalline structure emerge when the temperature is decreased? Dilute Particle Systems with Lennard-Jones potential Rome, University Tor Vergata, 7 November 200 Page 4 (4)

5 Assumptions on the Potential Assumption (V). v: [0, ) (, ] satisfies. There is ν 0 0 such that v = on [0,ν 0 ] and v < on (ν 0, ); 2. v is continuous on [0, ); 3. there is R > 0 such that v = 0 on [R, ); 4. there is ν > 0 such that v < 0 on (R ν,r); 5. there is ν 2 > 0 such that min v [0,ν 2 ] ν d 2 (2R) d sup s(r)r d min v. r (0,] [0, ) where s(r) denotes the minimal number of balls of radius r in R d required to cover a ball of radius one. In particular, v explodes at zero, v has a finite and strictly negative minimum, the support of v is bounded, 0 ν 0 ν 2 < R ν < R. Dilute Particle Systems with Lennard-Jones potential Rome, University Tor Vergata, 7 November 200 Page 5 (4)

6 The ground state (i.e., zero temperature): ϕ(n) = inf V N(x,...,x N ). x,...,x N R d Lemma. [STABILITY OF THE POTENTIAL] ϕ(n) ϕ = lim N N = inf ϕ(n) N N N (,0). Existence of limit by subadditivity, finiteness by Assumption (V)5., negativity by Assumption (V)4. The minimising configurations crystallise, i.e., approach a regular lattice (unique up to shift and rotation) in d = [GARDNER/RADIN 979] and in d = 2 [THEIL 2006]. Hence, the following sequence is continuous: { ϕ(κ) θ κ = κ, if κ N, ϕ, if κ =. Dilute Particle Systems with Lennard-Jones potential Rome, University Tor Vergata, 7 November 200 Page 6 (4)

7 The Limiting Free Energy Theorem. Fix c (0, ), then for any β n, exists and is given by Ξ(c) = lim logz N (β N,e cβ N ) N Nβ N q κ Ξ(c) = inf{ q κ θ κ c κ N { } κ N κ : q [0,]N { }, κ N { } } q κ =. Ξ(c) is the free energy per particle. In the case of fixed positive particle density at fixed positive temperature, the existence of the free energy per particle and of a close-packing phase transition are classical facts [RUELLE 999, Theorem 3.4.4]. Dilute Particle Systems with Lennard-Jones potential Rome, University Tor Vergata, 7 November 200 Page 7 (4)

8 Interpretation of the Formula Recall that the support of v is bounded by R. Hence, any point configuration {x,...,x N } decomposes into R-connected components. q κ is the relative frequency of the components of cardinality κ. More precisely: a given particle belongs with probability q κ to a component with κ elements. That is, {x,...,x N } consists of Nq κ /κ components of cardinality κ for each κ N (with a suitable adjustment for κ = ). Each component of cardinality κ is chosen optimally, i.e., as a minimiser in the definition of ϕ(κ). κ N { } q κ θ κ is the energy coming from such a configuration. c κ N q κ /κ is the entropy of the configuration (explanation follows). Neither information about the locations of the components relative to each other, nor about their shape is present. Dilute Particle Systems with Lennard-Jones potential Rome, University Tor Vergata, 7 November 200 Page 8 (4)

9 Analysis of the Formula Consider the sequence of points (/κ,θ κ ), κ N { }, and extend them to the graph of a piecewise linear function [0,] (,0]. Pick those of them which determine the largest convex minorant of this function, = κ < κ 2 <...: θ θ 3 c θ 4 θ 2 θ 6 7 c 2 θ 5 φ c 3 Put η = max{n N: κ n < } N { } and c n = θ κ n θ κn+, for n < η + /κ n /κ n+ Notation: q (κ) = (δ κ,n ) n N { } = κ-th unit sequence. Dilute Particle Systems with Lennard-Jones potential Rome, University Tor Vergata, 7 November 200 Page 9 (4)

10 The Phase Transitions Theorem 2. (i) The sequence (c n ) n<η+ is positive, finite and strictly decreasing. (ii) c, if c (c, ), ϕ(κ Ξ(c) = n ) κ n κ c n if c [c n,c n ) for some 2 n < η +, ϕ if c [0,c η ). (iii) For c (0, ) \ {c n : n < η + } the minimiser q is unique: for c (c, ) it is equal to q (κ ) = q (), for c (c n,c n ), with some 2 n < η +, it is equal to q (κn), for c = c it is equal to q ( ) (this is only applicable if η = and c > 0), for c (0,c η ) it is equal to q ( ). (iv) If c = c n for some n < η +, then the set of the minimisers is the set of convex combinations of certain q (i) s. η is the number of phase transitions. At least the high-temperature phase ( c < ) is non-empty, where the point configuration is totally disconnected. The low-temperature phase c is empty if η = and c η = 0. Dilute Particle Systems with Lennard-Jones potential Rome, University Tor Vergata, 7 November 200 Page 0 (4)

11 On the Proof: Empirical Measure Let x = {x,...,x N } be a configuration of points in Λ N, identified with its cloud N i= δ x i. It decomposes into its connected components [x i ] := j Θ i δ x j, Main object: the empirical measure on the connected components, translated such that any of its points is at the origin with equal measure: Y (x) N Then the energy is written N V N (x) = v ( x i x j ) N = i, j= i= i j = N Ψ ( ) Y (x) N, where = N N i= j i x j [x i ] δ [xi ] x i. v ( x i x j ) N = i= Ψ(Y) = Y(dA) #A v( x y ). x,y A x y #[x i ] x,y [x i ] x y v( x y ) Dilute Particle Systems with Lennard-Jones potential Rome, University Tor Vergata, 7 November 200 Page (4)

12 On the Proof: Large-Deviation Principle Let X be a vector of i.i.d. random variables X (N) Λ N, and write Y N = Y (X) N. Hence, (N),X 2,...,X (N) N Z N (β N,ρ N ) = Λ N N E ΛN [exp { β N Ψ(Y N ) }]. N! uniformly distributed on Proposition. (Y N ) N N satisfies a large-deviation principle with speed Nβ N and rate function [ J(Y) = c Y(dA) ]. #A That is, Nβ N logp ΛN ( YN ) = inf Y J(Y). Informally, Varadhan s lemma implies [ lim loge ΛN exp { β N Ψ(Y N ) }] { } = inf Ψ(Y)+J(Y). N Nβ N Y It is not difficult to see that this is basically Theorem. Dilute Particle Systems with Lennard-Jones potential Rome, University Tor Vergata, 7 November 200 Page 2 (4)

13 Example: More than one transition A one-dimensional example of a potential with η 2, i.e., at least two phase transitions: ν 0 R M M Potential v Phase transitions diagram (Satisfies Assumption (V).-5. with the exception of 4. A regularized version also satisfies 4.) Dilute Particle Systems with Lennard-Jones potential Rome, University Tor Vergata, 7 November 200 Page 3 (4)

14 Open Questions Analyse the precise size of the unbounded component(s). Does an unbounded support of v change anything? Add kinetic energy, i.e., consider the trace of exp{ β N H N }, where N H N = i + v( x i x j ). i= i< j N Non-dilute systems. Dilute Particle Systems with Lennard-Jones potential Rome, University Tor Vergata, 7 November 200 Page 4 (4)