Ginzburg-Landau Polynomials and the Asymptotic Behavior of the Magnetization in the Neighborhood of a Tricritical Point

Transcription

1 Richard S. Ellis, Asymptotic Behavior of the Magnetization Ginzburg-Landau Polynomials and the Asymptotic Behavior of the Magnetization in the Neighborhood of a Tricritical Point Richard S. Ellis Department of Mathematics and Statistics University of Massachusetts Amherst, MA 0003 Talk for the Workshop Dynamics and Thermodynamics of Systems with Long-Range Interactions: Theory and Experiments Assisi (Perugia), Italy 4-8 July 2007 Collaborators: Jonathan Machta Department of Physics University of Massachusetts Amherst Peter Tak-Hun Otto Department of Mathematics Willamette University Salem, OR rsellis@math.umass.edu

2 Richard S. Ellis, Asymptotic Behavior of the Magnetization 2 Outline of the Talk BEG Spin Model Phase transitions: second order (continuous bifurcation), first order (discontinuous bifurcation), tricritical point Main result Find x and θ such that magnetization m(β n, K n ) x/n θα 0, where (β n, K n ) converges to a second-order point or the tricritical point and α regulates the speed of approach. We work this out for 6 sequences. For each sequence, x is the unique, positive, global minimum point of the Ginzburg-Landau polynomial associated with the sequence. Ginzburg-Landau Polynomials Definition in terms of a limit of scaled free-energy functionals 8 applications. The G-L polynomials rigorize the G-L phenomenology of critical phenomena in the region of phase space through which (β n, K n ) passes. 2. The G-L polynomials reveal features of the phase-transition structure. 3. The asymptotic behavior of m(β n, K n ) 0 is expressed in terms of the unique, positive, global minimum point of G. 4. Scaling limit for total spin with limiting density const exp[ G]. 5. Moderate deviation principle (MDP) for total spin with rate function G const. 6. A consequence of the MDP led us to the asymptotic formula for m(β n, K n ). 7. MDP picks out physically relevant values of α in the asymptotic formula for m(β n, K n ). 8. Through m(β n, K n ) x/n θα, the G-L polynomials make rigorous Riedel s scaling approach to tricritical phase transitions. Contrast behavior near a second-order point versus near the tricritical point. We study 2 sequences (β n, K n ) converging to a second-order point. We study 4 sequences (β n, K n ) converging to the tricritical point. The behavior of the Ginzburg-Landau polynomials reveals the much more complicated phase transition structure near the tricritical point. References Phase Transitions in BEG Model: RSE, P. Otto, H. Touchette, AAP (2005) Limit Theorems in BEG Model: M. Costeniuc, RSE, P. T.-H. Otto, JSP (2007) Asymptotics of the Magnetization in the BEG Model: RSE, J. Machta, P. T.-H. Otto Scaling Approach to Tricritical Phase Transitions: E. K. Riedel, PRL (972)

3 Richard S. Ellis, Asymptotic Behavior of the Magnetization 3 Phase Transitions in the BEG Model K K(β) K(β) K(β ) c.0 K (β) β c β Figure : Sets describing phase-transition structure of the BEG model: the second-order curve {(β, K(β)), 0 < β < β c }, the first-order curve {(β, K (β)), β > β c }, and the tricritical point (β c, K(β c )). The phase-coexistence region consists of all (β, K) above the second-order curve, above the tricritical point, on the first-order curve, and above the first-order curve. The extension of the second-order curve to β > β c is called the spinodal curve. Define M β,k = the set of equilibrium values of the magnetization. Continuous bifurcation (second-order phase transition) for 0 < β < β c = log 4 as K K(β). Discontinuous bifurcation (first-order phase transition) for β > β c as K K (β). Tricritical point at (β c,k(β c )) separates the two phase transitions. Unique phase for (β, K) under the two phase-transition curves: M β,k = {0}. Double phase for (β, K) above the two phase-transition curves: M β,k = {±m(β,k)}, m(β,k) > 0. Triple phase for (β, K) on the first-order curve: M β,k = {±0,m(β,K)}, m(β,k) > 0.

4 Richard S. Ellis, Asymptotic Behavior of the Magnetization 4 Main Result (β n,k n ) (β,k(β)), 0 < β β c, second-order point or the tricritical point. α > 0 regulates speed of approach. Study fine structure of phase transitions. m(β n,k n ) 0. Plausible since M β,k = 0 for (β,k) = (β,k(β)). Find x and θ > 0 such that m(β n,k n ) x/n θα (n θα m(β n,k n ) x). K a 6 4b.09 K(β ) c.08 K(β) 4d.07.3 β c Figure 2: Sequences and 2 converge to a second-order point, sequences 3 6 converge to the tricritical point..4.5 K (β) 4c β 6 sequences (β n,k n ) with different asymptotic behavior of m(β n,k n ). Sequence : along ray above tangent line (K (β)b k < 0) β n = β + b/n α, K n = K(β) + k/n α 2. Sequence 2: along curve = second-order curve to order p 3. Sequence 3: along ray above tangent line (seq. for β = β c ) 4. Sequences 4a 4d: along curve tangent to spinodal curve (b > 0, l l c ) β n = β + b/n α, K n = K(β c ) + K (β c )b/n α + lb 2 /2n 2α 5. Sequence 5: along curve tangent to second-order curve (seq. 4 with b < 0, l > K (β c )) 6. Sequence 6: along curve = second-order curve to order p

5 Richard S. Ellis, Asymptotic Behavior of the Magnetization 5. Sequence : along ray above tangent line (K (β)b k < 0) β n = β + b/n α, K n = K(β) + k/n α 2. Sequence 2: along curve = second-order curve to order p (b 0, lb p > K (p) (β)b p ) [higher order correction to sequence ] β n = β + b/n α, K n = K(β) + p j= K(j) (β)b j /j!n jα + lb p /p!n pα 3. Sequence 3: along ray above tangent line (seq. for β = β c ) 4. Sequences 4a 4d: along curve tangent to spinodal curve (b > 0, l l c ) [higher order correction to sequence 3 for b > 0] β n = β + b/n α, K n = K(β c ) + K (β c )b/n α + lb 2 /2n 2α 5. Sequence 5: along curve tangent to second-order curve (seq. 4 with b < 0, l > K (β c )) 6. Sequence 6: along curve = second-order curve to order p (seq. 2 for β = β c with b < 0, lb p > K (p) (β)b p ) [higher order correction to sequence 3 for b < 0] Theorem. Asymptotic behavior of m(β n, K n ) for the 6 sequences. (a) Similar sequences give rise to different asymptotic behavior of m(β n,k n ) 0: compare and 3, 2 and 6 (influence of first-order curve). sequence converges to theorem asymptotic behavior in paper of m(β n, K n ) second-order point Thm. 3. m(β n,k n ) x/n α/2 2 second-order point Thm. 3.2 m(β n,k n ) x/n pα/2 3 tricritical point Thm. 6. m(β n,k n ) x/n α/4 4a 4d tricritical point Thm. 6.2 m(β n,k n ) x/n α/2 5 tricritical point Thm. 6.3 m(β n,k n ) x/n α/2 6 tricritical point Thm. 6.4 m(β n,k n ) x/n (p )α/2 (b) For each sequence (β n,k n ), x equals the unique, positive, global minimum point of the Ginzburg-Landau polynomial associated with (β n,k n ).

6 Richard S. Ellis, Asymptotic Behavior of the Magnetization 6 Ginzburg-Landau Polynomials Theorem. Free-energy functional and equilibrium values of the magnetization. Let G β,k denote the free-energy functional (see p. 8). Then M β,k = the set of global minimum points of G β,k. Definition of Ginzburg-Landau Polynomial G. Let α > 0 regulate the speed (β n,k n ) (β,k(β)) for 0 < β β c. Seek u R, γ > 0, and a polynomial G of degree 4 or 6 such that for all x R lim n u G βn,k n (x/n γ ) = G(x). n 8 Applications of the Ginzburg-Landau Polynomials. The Ginzburg-Landau polynomials G rigorize the Ginzburg-Landau phenomenology of critical phenomena in the region of phase space through which (β n,k n ) passes. 2. The Ginzburg-Landau polynomials G enable us to discover features of the phase-transition structure. 3. Asymptotic behavior of m(β n,k n ) is expressed in terms of the unique, positive, global minimum point x of G: m(β n,k n ) x/n θα. 4. Scaling limit with limiting density const exp[ G]: {S n /n γ dx} = exp[ G(x)]dx/ R exp[ G(y)]dy. 5. Moderate deviation principle (MDP) with rate function G const: {S n /n γ dx} exp[ n u (G(x) Ḡ)]dx, Ḡ = inf y R G(y). 6. We were led to the asymptotic formula m(β n,k n ) x/n θα by a consequence of the MDP: {S n /n γ dx} = ( 2 δ x + 2 δ x) (dx). 7. MDP picks out physically relevant values of α in the asymptotic formula m(β n,k n ) x/n θα. 8. Through m(β n,k n ) x/n θα, the Ginzburg-Landau polynomials make rigorous Riedel s scaling approach to tricritical phase transitions.

7 Richard S. Ellis, Asymptotic Behavior of the Magnetization 7 BEG Spin Model Mean-field approximation to Blume Capel model (966) and Blume Emery Griffiths model (97): study He 3 -He 4 mixtures and other physical systems. n spins ω i Λ = {,0,} 2. Microstates: ω = (ω,ω 2,...,ω n ) Λ n 3. Total spin: S n (ω) = n j= ω j 4. Hamiltonian or energy function (K > 0): H n,k (ω) = n j= ω2 j K n 5. Prior measure: n j,k= ω jω k = n j= ω2 j nk(s n(ω)/n) 2 P n (ω) = n j= ρ(ω j) = /3 n, ρ = 3 (δ + δ 0 + δ ) 6. Canonical ensemble (β > 0, K > 0): 7. Alignment effect: P n,β,k (ω) = Z n (β,k) exp[ βh n,k(ω)] /3 n, Z n (β,k) = ω Λ n exp[ βh n,k (ω)] /3 n where ρ β (ω j ) = e βω2 j /( + 2e β ), lim P n,β,k (ω) = P n (ω) = /3 n, β 0 + n lim P n,β,k (ω) = K 0 + j= ρ β(ω j ), lim P n,β,k (ω) = 2 (δ ω (ω) + δ ω +(ω)), K where for all j, ω j = and ω+ j = 8. Phase transition: persistence of alignment effect as n

8 Richard S. Ellis, Asymptotic Behavior of the Magnetization 8 Second-Order Phase Transition for β < β c Theorem. Large deviation principle and M β,k. (a) For β > 0, K > 0, we have the LDP with rate function I(β,K): P n,β,k {S n /n dx} exp[ ni β,k (x)]dx. (b) If I β,k (x) > 0, then P n,β,k {S n /n x} 0. (c) Define M β,k = {x R : I β,k (x) = 0} and the free-energy functional ( ) + e G β,k (x) = βkx 2 β (e 2βKx + e 2βKx ) log + 2e β = βkx 2 log Λ e2βkω ρ β (dω ) Then M β,k = {x R : x a global minimum point of G β,k } = M Gβ,K. (a) (b) Figure 3: Graphs of G β,k for 0 < β β c = log 4. (a) 0 < K K(β), (b) K > K(β) Theorem. Second-order phase transition for 0 < β β c = log 4. (a) For 0 < K K(β) = (e β + 2)/4β, M β,k = {0} (b) For K > K(β), m(β,k) > 0 such that M β,k = {±m(β,k)}. (c) m(β,k) 0 as K (K(β)) +. (d) M β,k has a continuous bifurcation at K = K(β) [second-order phase transition].

9 Richard S. Ellis, Asymptotic Behavior of the Magnetization 9 First-Order Phase Transition for β > β c (a) (b) Figure 4: Graphs of G β,k for β > β c. (a) 0 < K < κ(β), (b) κ(β) < K < K (β) Figure 5: Graph of G β,k for β > β c and K = K (β) (a) (b) Figure 6: Graphs of G β,k for β > β c. (a) K (β) < K < K(β), (b) K K(β) Theorem. First-order phase transition for β > β c = log 4. (a) For 0 < K < K (β), M β,k = {0}. (b) For K = K (β), m(β,k (β)) > 0 such that M β,k (β) = {0, ±m(β,k (β))}. (c) For K > K (β), m(β,k) > 0 such that M β,k = {±m(β,k)}. (d) m(β,k) m(β,k (β)) > 0 as K (K (β)) +. (e) M β,k has a discontinuous bifurcation at K = K (β) [first-order phase transition].

10 Richard S. Ellis, Asymptotic Behavior of the Magnetization 0 Ginzburg-Landau Phenomenology and Polynomials, I Phenomenology. For 0 < β < β c, K near K(β), and x near 0, c 4 (β) > 0 G β,k (x) G (2) β,k (0)x2 /2 + G (4) β,k (0)x4 /4! G β,k (x) = β(k(β) K)x 2 + c 4 (β)x 4. M Gβ,K has same form as M Gβ,K : point for 0 < K K(β), 2 points for K > K(β). Theorem. Ginzburg-Landau polynomial for sequence. For 0 < β < β c, α > 0, and K (β)b k < 0, define β n = β + b/n α, K n = K(β) + k/n α. (a) For u = 2α and γ = α/2 n u G βn,k n (x/n γ ) G(x) = β(k (β)b k)x 2 + c 4 (β)x 4, c 4 (β) > 0. (b) M G = {± x}, where x = β(k K (β)b)/2c 4 (β)). Proof. (a) Two-term Taylor expansion and asymptotics give Since n u G βn,k n (x/n γ ) n 2γ +uβ(k(β n) K n )x 2 + n 4γ +uc 4(β)x 4. K(β n ) K n = K(β + b/n α ) (K(β) + k/n α ) (K (β)b k)/n α, we have n u G βn,k n (x/n γ ) n 2γ+α +uβ(k (β)b k)x 2 + n 4γ +uc 4(β)x 4. Choose 2γ + α + u = 0 = 4γ + u or γ = α/2, u = 4γ = 2α, then n. (b) Solve G ( x) = 0.

11 Richard S. Ellis, Asymptotic Behavior of the Magnetization Ginzburg-Landau Polynomial Mirrors Phase-Transition Structure, I Theorem. Ginzburg-Landau polynomial for sequence. For 0 < β < β c, α > 0, and K (β)b k < 0, define β n = β + b/n α, K n = K(β) + k/n α. (a) For u = 2α and γ = α/2 n u G βn,k n (x/n γ ) G(x) = β(k (β)b k)x 2 + c 4 (β)x 4, c 4 (β) > 0. (b) M G = {± x}, where x = β(k K (β)b)/2c 4 (β)). (a) (b) Figure 7: Graphs of G β,k for 0 < β β c = log 4 and graphs of G. (a) Graph (a) shows the graph of G β,k for 0 < K K(β) and the graph of G for K (β)b k 0. (β n,k n ) (β,k(β)) from the single-phase region below the second-order curve. We have M Gβn,Kn = {0} = M G. (b) Graph (b) shows the graph of G β,k for K > K(β) and the graph of G for K (β)b k < 0. (β n,k n ) (β,k(β)) from the phase-coexistence region above the second-order curve. We have M Gβn,Kn = {±m(β n,k n )}, M G = {± x}, M Gβn,Kn = 2 = M G.

12 Richard S. Ellis, Asymptotic Behavior of the Magnetization 2 Asymptotics of m(β n, K n ) for Sequence Theorem. Ginzburg-Landau polynomial for sequence. For 0 < β < β c, α > 0, and K (β)b k < 0, define β n = β + b/n α, K n = K(β) + k/n α. (a) For u = 2α and γ = α/2 n u G βn,k n (x/n γ ) G(x) = β(k (β)b k)x 2 + c 4 (β)x 4, c 4 (β) > 0. (b) M G = {± x}, where x = β(k K (β)b)/2c 4 (β)). In the next theorem we give the asymptotics of m(β n,k n ) 0 for sequence. Theorem. Asymptotics of m(β n, K n ) for sequence. For 0 < β < β c, α > 0, and K (β)b k < 0, define Then β n = β + b/n α, K n = K(β) + k/n α. m(β n,k n ) x/n α/2. Proof. We write m n = n γ m(β n,k n ) and G n (x) = n u G βn,k n (x/n γ ). Since m(β n,k n ) is a global minimum point of G βn,k n, m n is a global minimum point of G n. We assert without proof that there exists x 0 such that m n x. Since G n ( m n ) G n (y) for all y and G n (x) G(x) for all x, G( x) = lim G n ( m n ) lim G n (y) = G(y). n n Thus x is a nonnegative global minimum point of G. It follows that x = x, the unique, nonnegative, global minimum point of G. Since γ = α/2, we obtain the desired conclusion: m n = n γ m(β n,k n ) = n α/2 m(β n,k n ) x; i.e.,m(β n,k n ) x/n α/2.

13 Richard S. Ellis, Asymptotic Behavior of the Magnetization 3 Ginzburg-Landau Phenomenology and Polynomials, II Phenomenology. For β > β c, β near β c, K near K(β c ), and x near 0, there exists c 4 > 0 and c 6 > 0 such that G β,k (x) G (2) β,k (0)x2 /2 + G (4) β,k (0)x4 /4! + G (6) β,k (0)x6 /6! G β,k (x) = β c (K(β) K)x 2 + c 4 (4 e β )x 4 + c 6 x 6. M G β,k has same form as M Gβ,K : point for 0 < K < K(β), 3 points for K = K (β), 2 points for K > K(β). K a 6 4b.09 K(β c).08 K(β) 4d.07 K (β) 4c.3 β c.4.5 β Theorem. Ginzburg-Landau polynomial for sequences 4a 4d. For β > β c, b > 0, α > 0, and l l c, define β n = β + b/n α, K n = K(β c ) + K (β)b/n α + lb 2 /2n 2 α. (a) For u = 3α and γ = α/2 n u G βn,k n (x/n γ ) G l (x) = 2 β c(k (β c ) l)b 2 x 2 4c 4 bx 4 + c 6 x 6. (b) For l = l c, M Glc = {0, ± x(l c )}. (c) For l > l c, M Gl = {± x(l)}. Proof. (a) Three-term Taylor expansion and asymptotics give n u G βn,k n (x/n γ ) n 2γ +uβ c(k(β n ) K n )x 2 + n 4γ +uc 4(4 e β n )x 4 + n 6γ c 6x 6. Since K(β n ) K n = K(β c + b/n α ) K n (K (β c ) l)b 2 /2n 2α, 4 e β n 4b/n α, n u G βn,k n (x/n γ ) n β 2γ+2α +u 2 c(k (β c ) l)x 2 n 4γ+α +u4c 4bx 4 + n 6γ +uc 6x 6. Choose 2γ + 2α + u = 0 = 4γ + α + u = 6γ + u or γ = α/2, u = 6γ = 3α, then n. (b) (c) Solve G l ( x) = 0.

14 Richard S. Ellis, Asymptotic Behavior of the Magnetization 4 Ginzburg-Landau Polynomial Mirrors Phase-Transition Structure, II Theorem. Ginzburg-Landau polynomial G l for sequences 4a 4d. For β > β c, b > 0, α > 0, and l l c = K (β c ) 5/4β c, define β n = β + b/n α, K n = K(β c ) + K (β)/n α + lb 2 /2n 2 α. (a) For u = 3α and γ = α/2 n u G βn,k n (x/n γ ) G l (x) = 2 β c(k (β c ) l)b 2 x 2 4c 4 bx 4 + c 6 x 6. (b) For l = l c, M Glc = {0, ± x(l c )}. (c) For l > l c, M Gl = {± x(l)}. Since β n β c = b/n α, (β n,k n ) (β c,k(β c )) along the curve (β, K(β)), where for β > β c K(β) = K(β c ) + K (β c )(β β c ) + l(β β c ) 2 /2. K a 6 4b.09 K(β c).08 K(β) 4d.07 K (β) 4c.3 β c.4.5 β (a) Sequence 4a corresponds to l > K (β c ). (b) Sequence 4b corresponds to l = K (β c ). (c) Sequence 4c corresponds to l = l c = K (β c ) 5/4β c ; G lc has 3 global minimum points. (d) Sequence 4d corresponds to l c < l < K (β c ).

15 Richard S. Ellis, Asymptotic Behavior of the Magnetization 5 (a) (b) Figure 8: Graphs of G β,k for β > β c, 0 < K < K (β) and graphs of G l for l < l c. Figure 9: Graph of G β,k for β > β c, K = K (β) and graph of G l for l = l c. (a) (b) Figure 0: Graphs of G β,k for β > β c, K > K (β) and graphs of G l for l > l c.. Figure 9. For l = l c, M Glc = {0, ± x(l c )}. We argue that (β n,k n ) (β c,k(β c )) along (β, K(β)), which coincides with first-order curve to order 2. This implies K (β c) = l c. M Glc = {0, ± x(l c )} mirrors M = {0, ±m(β Gβn,Kn n,k n )} for (β n,k n ) on first-order curve. 2. Figure 8. For l < l c, (β n,k n ) (β c,k(β c )) from the single-phase region under first-order curve. We have M Gβn,Kn = {0} = M G l. 3. Figure 0. For l > l c, (β n,k n ) (β c,k(β c )) from the phase-coexistence region above first-order curve. We have M Gβn,Kn = {±m(β n,k n )}, M Gl = {± x(l)}, M Gβn,Kn = 2 = M G l. 4. M Gl undergoes a first-order phase transition at l = l c, mirroring the first-order phase transition in M Gβn,Kn at K = K (β n ) for β n > β c.

16 Richard S. Ellis, Asymptotic Behavior of the Magnetization 6 Asymptotics of m(β n, K n ) for Sequences 4a 4d Theorem. Ginzburg-Landau polynomial for sequences 4a 4d. For β > β c, b > 0, α > 0, and l l c, define β n = β + b/n α, K n = K(β c ) + K (β)b/n α + lb 2 /2n 2 α. (a) For u = 3α and γ = α/2 n u G βn,k n (x/n γ ) G l (x) = 2 β c(k (β c ) l)b 2 x 2 4c 4 bx 4 + c 6 x 6. (b) For l = l c, M Glc = {0, ± x(l c )}. (c) For l > l c, M Gl = {± x(l)}. In the next theorem we give the asymptotics of m(β n,k n ) 0 for sequences 4a 4d. These are the only sequences for which higher order corrections do not affect the asymptotic behavior of m(β n,k n ). Theorem. Asymptotics of m(β n, K n ) for sequence 4a 4d. For β > β c, b > 0, α > 0, and l l c, let (β n,k n ) be the sequence in the preceding theorem. Then m(β n,k n ) x(l)/n α/2. Proof. For l > l c, the proof is the same as the proof for sequence (see p. 0). The proof for l = l c is much harder. We write m n = n γ m(β n, K n ) and G n (x) = n u G βn,k n (x/n γ ). Since m(β n, K n ) is a global minimum point of G βn,k n, m n is a global minimum point of G n. We assert without proof that there exists x 0 such that m n x. Since G n ( m n ) G n (y) for all y and G n (x) G lc (x) for all x, G lc ( x) = lim G n ( m n ) lim G n (y) = G lc (y). n n Thus x is a nonnegative global minimum point of G lc. In the proof for sequence, we were able to assert that x = x, the unique, nonnegative, global minimum point of G. However, G lc has two nonnegative global minimum points, 0 and x(l c ). If x = 0, then we do not obtain the desired asymptotic formula. In order to prove that x = x(l c ), one must use results from complex analysis, including Rouché s theorem. After one shows that x = x(l c ), the proof is completed as before. Since γ = α/2, the desired conclusion follows: m n = n γ m(β n, K n ) = n α/2 m(β n, K n ) x(l c ); i.e., m(β n, K n ) x(l c )/n α/2.

17 Richard S. Ellis, Asymptotic Behavior of the Magnetization 7 Ginzburg-Landau Polynomial and Scaling Limit Theorem. Ginzburg-Landau polynomial for sequence. For 0 < β < β c, α > 0, and K (β)b k < 0, define β n = β + b/n α, K n = K(β) + k/n α. (a) For u = 2α and γ = α/2 n u G βn,k n (x/n γ ) G(x) = β(k (β)b k)x 2 + c 4 (β)x 4, c 4 (β) > 0. (b) M G = {± x}, where x = β(k K (β)b)/2c 4 (β)). Theorem. Scaling limit for S n /n 3/4 for sequence. For 0 < β < β c, α > 0, and K (β)b k < 0, let (β n,k n ) be the sequence in the preceding theorem. Then {S n /n 3/4 dx} = exp[ G(x)]dx/ R exp[ G(y)]dy; i.e., for any bounded, continuous function f lim n Λ n f(s n/n 3/4 ) d = R f(x) exp[ G(x)]dx/ R exp[ G(y)]dy. Proof. In part (a) of the first theorem on this page, set u = 0. Then for γ = /4 ng βn,k n (x/n γ ) = ng βn,k n (x/n /4 ) G(x) = β(k (β)b k)x 2 + c 4 (β)x 4, c 4 (β). Omitting several technical steps, we have for any bounded, continuous function f lim n Λ n f(s n/n 3/4 )d = lim f(x) exp[ ng β n R n,k n (x/n /4 )] dx/ exp[ ng β R n,k n (y/n /4 )] dy = R f(x)exp[ G(x)] dx/ exp[ G(y)] dy. R This completes the proof. The first equality is a consequence of the Hubbard-Stratonovitch transformation.

18 Richard S. Ellis, Asymptotic Behavior of the Magnetization 8 Ginzburg-Landau Polynomial and Moderate Deviation Principle Theorem. Ginzburg-Landau polynomial for sequence. For 0 < β < β c, α > 0, and K (β)b k < 0, define β n = β + b/n α, K n = K(β) + k/n α. (a) For u = 2α = 4γ and γ = α/2 n u G βn,k n (x/n γ ) G(x) = β(k (β)b k)x 2 + c 4 (β)x 4, c 4 (β) > 0. (b) M G = {± x}, where x = β(k K (β)b)/2c 4 (β)). The weak limit in part (b) of the next theorem led us to the asymptotic formula for m(β n,k n ) in the theorem on page 2. Theorem. Moderate deviation principle for S n /n γ for sequence. For 0 < β < β c, γ (0,/4), α = 2γ (0,/2), and K (β)b k < 0, let (β n,k n ) be the sequence in the preceding theorem. (a) Define Γ(x) = G(x) min y R G(y). We have the MDP {S n /n γ dx} exp[ n 4γ Γ(x)]dx, (b) We have the weak limit {S n /n γ dx} = ( 2 δ x + 2 δ x) (dx). Definition of MDP and Laplace Principle. If A is any set in R, then Γ(A) denotes the infimum of Γ over A. The MDP states for every closed set F in R lim sup n n log P 4γ n,β n,k n {S n /n γ F } Γ(F) and that for every open set U in R lim inf n n log P 4γ n,β n,k n {S n /n γ U} Γ(U). The MDP is equivalent to the Laplace principle, which states that for any bounded, continuous function ψ lim n n 4γ Λ n exp[ n 4γ ψ(s n /n γ )]d = sup{ψ(x) Γ(x)}. x R

19 Richard S. Ellis, Asymptotic Behavior of the Magnetization 9 Proof of the Laplace principle. We prove that for any bounded, continuous function ψ {ψ(x) Γ(x)}. n 4γ lim n Λ n exp[ n 4γ ψ(s n /n γ )] d = sup x R According to part (a) of the theorem on page 8, for u = 2α = 4γ and γ = α/2 n u G βn,k n (x/n γ ) = n 4γ G βn,k n (x/n γ ) G(x) = β(k (β)b k)x 2 + c 4 (β)x 4, c 4 (β) > 0. For the Laplace principle and thus the MDP, we require u = 4γ > 0, which restricts γ and α = 2γ to the respective open intervals (0, /4) and (0, /2). Omitting several technical steps, we have for any bounded, continuous function ψ Λ n exp[ n 4γ ψ(s n /n γ )] d R exp[n 4γ {ψ(x) n 4γ G βn,k n (x/n γ )}] dx/ R exp[n 4γ {ψ(x) G(x)}] dx/ R exp[n 4γ { G(x)}] dx exp[n 4γ sup x R {ψ(x) G(x)}] / exp[ n 4γ inf x R {G(x)}] = exp[n 4γ sup x R {ψ(x) Γ(x)}]. R exp[n 4γ { n 4γ G βn,k n (x/n γ )}] dx The next to last line is a consequence of Laplace s method for the asymptotics of integrals over R. The limit giving the Laplace principle follows from this display. Proof of weak limit in part (b). We have G(± x) = min y R G(y) and Γ(± x) = 0 < Γ(y) for all y ± x. Let F be any closed set in R not containing ± x. Then Γ(F) > 0, and the moderate deviation upper bound implies that for all sufficiently large n {S n /n γ F } exp[ n 4γ Γ(F)] 0. A separate argument shows that the weak limit of the sequence {S n /n γ dx} exists. The last display implies that this weak limit equals ( 2 δ x + 2 δ x)(dx). Weak limit in part (b) led us to asymptotic formula for m(β n, K n ). According to the theorem on page 2, m(β n, K n ) x/n α/2 for any α > 0. To derive this asymptotic formula, we start with the heuristic formula {S n /n dx} ( 2 δ m(β n,k n) + 2 δ m(β n,k n))(dx), which expresses the fact that ±m(β n, K n ) are the equilibrium values of S n /n. Replace dx with dx/n γ and obtain {S n /n γ dx} ( 2 δ n γ m(β n,k n) + 2 δ n γ m(β n,k n))(dx). Compare this with the weak limit in part (b), which holds for α (0, /2): ( {S n /n γ dx} = ) 2 δ x + δ 2 x (dx). This leads to the desired asymptotic formula for α (0, /2): n γ m(β n, K n ) x(l) or m(β n, K n ) x/n α/2.

20 Richard S. Ellis, Asymptotic Behavior of the Magnetization 20 Ginzburg-Landau Polynomial and Moderate Deviation Principle Theorem. Ginzburg-Landau polynomial for sequence. For 0 < β < β c, α > 0, and K (β)b k < 0, define β n = β + b/n α, K n = K(β) + k/n α. (a) For u = 2α = 4γ and γ = α/2 n u G βn,k n (x/n γ ) G(x) = β(k (β)b k)x 2 + c 4 (β)x 4, c 4 (β) > 0. (b) M G = {± x}, where x = β(k K (β)b)/2c 4 (β)). The weak limit in part (b) of the next theorem led us to the asymptotic formula for m(β n,k n ) in the theorem on page 2. Theorem. Moderate deviation principle for S n /n γ for sequence. For 0 < β < β c, γ (0,/4), α = 2γ (0,/2), and K (β)b k < 0, let (β n,k n ) be the sequence in the preceding theorem: β n = β + b/n α, K n = K(β) + k/n α. (a) Define Γ(x) = G(x) inf y R G(y). We have the MDP {S n /n γ dx} exp[ n 4γ Γ(x)]dx, (b) We have the weak limit {S n /n γ dx} = ( 2 δ x + 2 δ x) (dx). Definition of MDP. If A is any set in R, then Γ(A) denotes the infimum of Γ over A. The MDP states for every closed set F in R lim sup n and that for every open set U in R lim inf n n 4γ log P n,β n,k n {S n /n γ F } Γ(F) n 4γ log P n,β n,k n {S n /n γ U} Γ(U).

21 Richard S. Ellis, Asymptotic Behavior of the Magnetization 2 Theorem. Moderate deviation principle for S n /n γ for sequence. For 0 < β < β c, γ (0,/4), α = 2γ (0,/2), and K (β)b k < 0, define β n = β + b/n α, K n = K(β) + k/n α and the Ginzburg-Landau polynomial G(x) = β(k (β)b k)x 2 + c 4 (β)x 4. (a) Define Γ(x) = G(x) inf y R G(y). We have the MDP {S n /n γ dx} exp[ n 4γ Γ(x)]dx, (b) We have the weak limit {S n /n γ dx} = ( 2 δ x + 2 δ x) (dx). Proof of weak limit in part (b). We have G(± x) = min y R G(y) and Γ(± x) = 0 < Γ(y) for all y ± x. Let F be any closed set in R not containing ± x. Then Γ(F) > 0, and for all sufficiently large n {S n /n γ F } exp[ n 4γ Γ(F)] 0. A separate argument shows that the weak limit of the sequence {S n /n γ dx} exists. The last display implies that this weak limit equals ( 2 δ x + 2 δ x)(dx). Weak limit in part (b) led us to asymptotic formula for m(β n, K n ). Theorem on page 2: for any α > 0,m(β n,k n ) x/n α/2. Since ±m(β n,k n ) = equilibrium values of S n /n, we have heuristic formula {S n /n dx} ( 2 δ m(β n,k n ) + 2 δ m(β n,k n ))(dx), Replace dx with dx/n γ and obtain {S n /n γ dx} ( 2 δ n γ m(β n,k n ) + 2 δ n γ m(β n,k n ))(dx). Compare with weak limit in part (b), which holds for α (0, /2): {S n /n γ dx} = ( 2 δ x + 2 δ x) (dx). This leads to n γ m(β n,k n ) x or m(β n,k n ) x/n α/2 for α (0,/2). Note different ranges of α. Here α (0,/2) versus any α > 0 in the theorem on page 2. The MDP is telling us something physically important.

22 Richard S. Ellis, Asymptotic Behavior of the Magnetization 22 MDP Picks Out Physically Relevant Values of α For 0 < β < β c, γ (0,/4), α = 2γ (0,/2), K (β)b k < 0, define β n = β + b/n α and K n = K(β) + k/n α. Case. α > /2. See figure (a). (β n,k n ) (β,k(β)) quickly, and we are in the critical regime, where finite-size scaling matters. m(β n,k n ), the positive global minimum point of G βn,k n, is much smaller than the fluctuations of S n /n: m(β n,k n ) x/n α/2 E n,βn,k n S n /n const/n /4. m(β n,k n ) has no physical meaning. S n /n has large fluctuations around 0. (a) (b) Gβ n, K n P n, β n, K( S dx) n n/n - m( β n, K n ) m( β n, K n ) - m( β n, K n ) m( β n, K n ) Figure : G βn,k n and {S n /n dx}. (a) α /2, (b) 0 < α < /2 Case 2. 0 < α < /2 (values of α for MDP). See figure (b). (β n,k n ) (β,k(β)) slowly, and we are in the two-phase regime, where the system is effectively infinite. G βn,k n has two, deep, global minimum points at ±m(β n,k n ), and the probability distribution of S n /n is sharply peaked at ±m(β n,k n ): m n x n α/2 E n,βn,k n { Sn /n m n Sn /n > δm n } const, δ (0,). n /2 α/2 E n,βn,k n S n /n m(β n,k n ) x/n α/2. m(β n,k n ) has physical meaning because it is the expected value of a measurement of S n /n.

23 Richard S. Ellis, Asymptotic Behavior of the Magnetization 23 Scaling Approach to Tricritical Phase Transitions K a 6 4b.09 K(β c).08 K(β) 4d.07 K (β) 4c.3 β c.4.5 β sequence converges to theorem asymptotic behavior in paper of m(β n, K n ) second-order point Thm. 3. m(β n, K n ) x/n α/2 2 second-order point Thm. 3.2 m(β n, K n ) x/n pα/2 3 tricritical point Thm. 6. m(β n, K n ) x/n α/4 4a 4d tricritical point Thm. 6.2 m(β n, K n ) x/n α/2 5 tricritical point Thm. 6.3 m(β n, K n ) x/n α/2 6 tricritical point Thm. 6.4 m(β n, K n ) x/n (p )α/2 Connection with scaling theory Sequence. /n α = distance from (β n,k n ) to second-order curve, and critical exponent for m(β n,k n ) is /2. Thus m(β n,k n ) 0 like /n α/2. This is the case p = 2 of sequence 2. Sequence 2. /n pα = distance from (β n,k n ) to second-order curve, and critical exponent for m(β n,k n ) is /2. Thus m(β n,k n ) 0 like /n pα/2. Sequence 3. /n α = distance from (β n,k n ) to first-order curve, and critical exponent for m(β n,k n ) is /4. Thus m(β n,k n ) 0 like /n α/4. Sequences 4a 4d. This can be seen via Riedel s scaling approach to tricritical phase transitions for region I in his paper. Sequence 5. This is the case p = 2 of sequence 6. Sequence 6. This can be seen via Riedel s scaling approach to tricritical phase transitions for region II in his paper.

24 Richard S. Ellis, Asymptotic Behavior of the Magnetization 24 Summary of Talk Theorem. Free-energy functional and equilibrium values of the magnetization. Let G β,k denote the free-energy functional (see p. 8). Then M β,k = the set of global minimum points of G β,k. Definition of Ginzburg-Landau Polynomial G. Let α > 0 regulate the speed (β n,k n ) (β,k(β)) for 0 < β β c. Seek u R, γ > 0, and a polynomial G of degree 4 or 6 such that for all x R lim n u G βn,k n (x/n γ ) = G(x). n 8 Applications of the Ginzburg-Landau Polynomials. The Ginzburg-Landau polynomials G rigorize the Ginzburg-Landau phenomenology of critical phenomena in the region of phase space through which (β n,k n ) passes. 2. The Ginzburg-Landau polynomials G enable us to discover features of the phase-transition structure. 3. Asymptotic behavior of m(β n,k n ) is expressed in terms of the unique, positive, global minimum point x of G: m(β n,k n ) x/n θα. 4. Scaling limit with limiting density const exp[ G]: {S n /n γ dx} = exp[ G(x)]dx/ R exp[ G(y)]dy. 5. Moderate deviation principle (MDP) with rate function G const: {S n /n γ dx} exp[ n u (G(x) Ḡ)]dx, Ḡ = inf y R G(y). 6. We were led to the asymptotic formula m(β n,k n ) x/n θα by a consequence of the MDP: {S n /n γ dx} = ( 2 δ x + 2 δ x) (dx). 7. MDP picks out physically relevant values of α in the asymptotic formula m(β n,k n ) x/n θα. 8. Through m(β n,k n ) x/n θα, the Ginzburg-Landau polynomials make rigorous Riedel s scaling approach to tricritical phase transitions.