A simple spin model with a nonconcave entropy
Hugo Touchette
School of Mathematical Sciences, Queen Mary, University of London
Padova, November 21, 2006
Supported by NSERC (Canada), the Royal Society, and HEFCE (England)
Hugo Touchette (QMUL) Nonconcave entropy November 21, 2006 1 / 19
Outline
1 Revision of concepts
2 Spin model with nonconcave entropy
3 Generalized canonical ensembles
4 Conclusion
Hugo Touchette (QMUL) Nonconcave entropy November 21, 2006 2 / 19 Microcanonical ensemble
Ωu Ω microstate space Ω u = U/N mean energy U Ωu energy shell P = const for microstate ∈ Ωu
Density of states:
ρ(u) = #microstates with mean energy u
Microcanonical entropy function: 1 s(u) = lim ln ρ(u) N→∞ N
Hugo Touchette (QMUL) Nonconcave entropy November 21, 2006 3 / 19
Canonical ensemble
1 e−βU β = = const, P = kB T Z(β)
A B Canonical partition function: X Z(β) = e−βU microstates Canonical free energy function: 1 ϕ(β) = lim − ln Z(β) N→∞ N
Hugo Touchette (QMUL) Nonconcave entropy November 21, 2006 4 / 19 Basic properties
Free energy ϕ(β) is always concave ϕ(β) is the Legendre transform of s(u)
0 ϕ(β) = βuβ − s(uβ), s (uβ) = β
Entropy
s(u) is always concave s(u) is the Legendre transform of ϕ(β)
0 Not always true! s(u) = βuu − ϕ(βu), ϕ (βu) = u
The microcanonical and canonical ensembles are always equivalent
Hugo Touchette (QMUL) Nonconcave entropy November 21, 2006 5 / 19
Block spin model Touchette, cond-mat/0504020
↑ ↓ ··· ↑ ↑ ↑ ... ↑ ↑ H
N free spins N correlated (frozen) spins
N N X X Ufree = si Ucorr = si = Nsi i=1 i=1
si = ±1, µH = 1 si = ±1 Total energy: Utot = Ufree + Ucorr Mean energy: U u = tot ∈ [−2, 2] N
Hugo Touchette (QMUL) Nonconcave entropy November 21, 2006 6 / 19 Microcanonical entropy
M1 M2 ln2 0.6 G: ↓↓ · · · ↓ | ↓↓ · · · ↓ M1 : ↓↑ · · · ↓ | ↓↓ · · · ↓ 0.4 s(u) M2 : ↓↑ · · · ↓ | ↑↑ · · · ↑ C: ↑↑ · · · ↑ | ↓↓ · · · ↓ 0.2 ↓↓ · · · ↓ | ↑↑ · · · ↑ 0 G C E E: ↑↑ · · · ↑ | ↑↑ · · · ↑ - 2 - 1 0 1 2 u
s (u + 1) if u ∈ [−2, 0] s(u) = 0 s0(u − 1) if u ∈ (0, 2], 1 − u 1 − u 1 + u 1 + u s (u) = − ln − ln 0 2 2 2 2
Hugo Touchette (QMUL) Nonconcave entropy November 21, 2006 7 / 19
Canonical free energy
Solution: ϕ(β) = − ln(2 cosh β) − |β| Non-differentiable at β = 0 First-order phase transition in canonical ensemble
Hugo Touchette (QMUL) Nonconcave entropy November 21, 2006 8 / 19 Equilibrium mean energy in canonical ensemble
2 0 uβ = ϕ (β) 1 ↓↑ · · · ↓ | ↑↑ · · · ↑ u = +1
uβ 0
-1 ↓
-2 ↓↑ · · · ↓ | ↓↓ · · · ↓ u = −1 -4 -2 0 2 4 β First-order phase transition at β = 0 Canonical “does not see” the mean energy region u ∈ (−1, 1) Nonequivalence of ensembles for u ∈ (−1, 1)
Hugo Touchette (QMUL) Nonconcave entropy November 21, 2006 9 / 19
Nonequivalence of ensembles
0.6
0.4 s(u)
0.2
0 - 2 - 1 0 1 2 u
ϕ = s∗ always s = ϕ∗ for u 6∈ (−1, 1) s 6= ϕ∗ for u ∈ (−1, 1)
Hugo Touchette (QMUL) Nonconcave entropy November 21, 2006 10 / 19 Generalized canonical ensemble
Costeniuc, Ellis, Touchette & Turkington JSP 119, 1283 (2005) PRE 73, 026105 (2006) Energy: U Mean energy: u = U/N
Standard canonical Generalized canonical e−Nβu e−Nαu−Ng(u) Pβ = Pg,α = Z(β) Zg (α)
X −Nβu X −Nαu−Ng(u) Z(β) = e Zg (α) = e microstates microstates 1 1 ϕ(β) = lim − ln Z(β) ϕg (α) = lim − ln Zg (α) N→∞ N N→∞ N
Hugo Touchette (QMUL) Nonconcave entropy November 21, 2006 11 / 19
Gaussian ensemble
Challa & Hetherington (1988) Pivot function: g(u) = γu2 Gaussian Gibbs measure:
e−Nαu−Nγu2 Pγ,α = Zγ(α) Gaussian partition function:
X −Nαu−Nγu2 Zγ(α) = e microstates Gaussian free energy: 1 ϕγ(α) = lim − ln Zγ(α) N→∞ N
Hugo Touchette (QMUL) Nonconcave entropy November 21, 2006 12 / 19 Gaussian free energy
0
-2 γ = 10 -4 ϕγ(α) γ = 2 -6
-8 γ = 0 -10 -4 -2 0 2 4 α
Nondifferentiable point disappears as γ → ∞ Closing of the mean energy gap
Hugo Touchette (QMUL) Nonconcave entropy November 21, 2006 13 / 19
Equilibrium mean energy
2
1
u γ ,α 0
-1
γ = 0 γ = 2 γ = 10 -2 -30 -20 -10 0 10 20 30 α
0 Equilibrium mean energy: uγ,α = ϕγ(α) Closing of the mean energy gap First-order phase transition disappears as γ → ∞
Hugo Touchette (QMUL) Nonconcave entropy November 21, 2006 14 / 19 Recovering the entropy
γ = 0 γ = 2 0
-2 γ = 10 γ = 10 -4 s(u) ϕγ(α) γ = 2 -6
-8 0 γ = ∞ γ = -10 -4 -2 0 2 4 α -2 -1 0 1 2 u Calculation of the entropy:
2 0 s(u) = αuu − ϕγ(αu) +γu , ϕγ(αu) = u
| {z } ∗ Legendre transform: ϕγ
Correct entropy recovered as γ → ∞
Hugo Touchette (QMUL) Nonconcave entropy November 21, 2006 15 / 19
Conclusion
s is not necessarily concave If s is nonconcave, then s 6= ϕ∗ Basis for nonequivalence of ensembles Related to first-order phase transitions
Generalized canonical free energy: ϕ → ϕg ∗ Recovering equivalence: s = ϕg + g
Hugo Touchette (QMUL) Nonconcave entropy November 21, 2006 16 / 19 Systems with nonconcave entropies
Gravitating particles (Thirring, Lynden-Bell, 1970s) Mean-field and long-range spin models
I Blume-Emery-Griffiths model I Potts model (q > 2) 4 I Hamiltonian mean-field model (φ ) Model of plasma Statistical models of turbulence Typically: systems with long-range potentials
Hugo Touchette (QMUL) Nonconcave entropy November 21, 2006 17 / 19
Other applications Multifractals Nonconcave multifractal spectrum:
−f (α) nε(α) ∼ ε
Touchette & Beck JSP (2006)
Nonequilibrium fluctuation theorems Nonconvex fluctuation functions:
−tf (w) P(Wt = w) ∼ e
Large deviation theory Nonconvex rate functions:
−nI (a) P(An = a) ∼ e
Hugo Touchette (QMUL) Nonconcave entropy November 21, 2006 18 / 19 General references
T. Dauxois, S. Ruffo, E. Arimondo, and M. Wilkens Dynamics and Thermodynamics of Systems with Long Range Interactions Springer Verlag, New York, 2002. H. Touchette, R.S. Ellis, and B. Turkington An introduction to the thermodynamic and macrostate levels of nonequivalent ensembles Physica A 340, 138, 2004.
http://www.maths.qmul.ac.uk/˜ht [email protected]
Hugo Touchette (QMUL) Nonconcave entropy November 21, 2006 19 / 19