Statistical Thermodynamics: Molecules to Machines

Statistical Thermodynamics: Molecules to MachinesVenkat Viswanathan May 20, 2015Module 5: Ising modelLearning Objectives: Introduce the Ising model for magnetic systems. Exactly solve the Ising model in 1 dimension. Give the mean-field approximation for the Ising model in D 2. Discuss critical behavior in magnetic systems and in liquid-vapor systems Show how the mean-field approximation does not accurately ac- count for correlations that dominate the behavior near the critical point.Key Concepts:Ising model, dimensionality, ferromagnetic and antiferromagnetic, mag- netization, transfer function, transfer matrix, meanfield approximation, critical behavior, fluctuations, critical fluctuations, Rayleigh scatteringNon-interacting moleculesThus far, our analyses have focused on non-interacting systems: Ideal gas Indistinguishable, non-interacting bosons Electron gas Indistinguishable, non-interacting fermions Crystal fluctuations Distinguishable, non-interacting phonons (quasi- particles).Non-interacting systems require the evaluation of a single-molecule partition function, which generally is tractable (either exactly or ap- proximately). However, very few practical problems involve either non- interacting molecules or approximately non-interacting molecules. Fur- thermore, a variety of important physical phenomena are not exhibited in non-interacting systems. For example, phase transitions are a hall- mark issue in interacting systems. Our goal for the remainder of this class is to discuss the effect of interactions on thermodynamic behavior.Ising modelConsider a lattice of N sites. The definition of the lattice can include many different geometries (square lattice, triangular lattice, etc..) and dimensionality (1 dimension, 2 dimensions, etc...). Each site contains exactly 1 spin that is defined by the spin state si that is either spin up(si = +1) or spin down (si = 1). The lattice spins are subjected to anexternal field h, and the spins only interact with there nearest neighbors with a coupling strength J . The system energy is defined to be:NE = h . si J . sisj(1)i=1

(ij)where the sum over (ij) implies a summation over all nearest neighborpairs in the lattice.This general model can be applied to a variety of problems (magnetic systems, binary alloys, liquid-gas phase transition, neuron network - Hopfield model).The sign of the coupling constant J dictates whether the spins prefer to align with their neighbors or anti-align (Fig. 1): Ferromagnetic the spins prefer to align with their neighbors, occur- ring when J > 0. Antiferromagnetic the spins prefer to anti-align with their neighbors, occurring when J < 0.

Figure 1: Ground states (minimum en- ergy) for the 2-dimensional Ising model (square lattice) for a ferromagnetic sys- tem (with coupling constant J > 0) and an antiferromagnetic system (with coupling constant J < 0).Non-interacting case (J = 0)The partition function for the Ising model with J = 0 is given by.N.Q =.

. ....

exp

h . sis1 =1,1 s2 =1,1

sN =1,1

i=1= .s1 =1,1

ehs1 .s2 =1,1N

ehs2 ... .sN =1,1N

ehsN = .s=1,1

ehs

= .eh + eh.

= 2N coshN (h)which is valid for any lattice type and dimensionality.Define the total magnetization M to be:. N.M =. si=i=1

log Q

h

= N tanh (h)(2)The average energy (E) and entropy S are given by:(E) = h

. N.. sii=1

= hM = hN tanh (h)S = E FT

= hN tanh (h) + NkBT log [2 cosh (h)]TNote, (E) hN and S 0 as T 0, and (E) 0 and S NkB log 2 as T .The non-interacting Ising model (J = 0) exhibits a gradual changein the magnetization M with the external field h (Fig. 2). Thus, no phase transition is exhibited in the non-interacting case; there is never a phase transition for non-interacting molecules. We now turn to caseswhere the interactions are turned on, such that J = 0.Ising model in 1 dimensionConsider a line of spins in 1 dimension with periodic boundary conditions such that sN interacts with s1. For the Ising model in 1 dimension, the energy is written as:

Figure 2: The total magnetization M = N versus the external field h for the non-interacting Ising model J = 0).NNE = h . si J . sisi+1(3)where h = h and J = J .

i=1

i=1Define a transfer function T (si, si+1) to be:T (si, si+1) = exp .h .2

+ si+12

. + Jsisi+1.(4)With this definition, we write the partition function as:Q =.

. ....

T (s1, s2)T (s2, s3)...T (sN , s1)s1 =1,1 s2 =1,1sN =1,1Now construct the transfer matrix to be:. T (1, 1)T (1, 1)T =T (1, 1)T (1, 1)

.. eh +J=eJ

eJ.eh +J

(5)The partition function is then written as:Q =.s1 =1,1

TN (s1, s1) = Tr{TN }(6)where Tr{TN } indicates a trace of the matrix M .The trace of a matrix is the sum of its eigenvalues, and the eigenvaluesof TN are N , where are the eigenvalues of T . In our case: = eJ

cosh h

.e

2J

+ e2J

sinh

2 h .1/2

(7)Therefore, the exact solution for the partition function is:Q = eJ cosh h + .e2J + e2J sinh2 h .1/2. +eJ cosh h .e2J + e2J sinh2 h .

1/2.NFor J = 0, the partition function Q = 2N coshN h , which agrees with our previous result. For h = 0, the partition function Q = 2N coshN J +2N sinhN J 2N coshN J for large N .Since + > , the partition function is Q N

for N 1. Theaverage magnetization is given by M = log Q

= N

sinh he4J +sinh2 h

. Inthe limit h 0+, the magnetization approaches zero. Thus, this exactsolution for the 1D Ising model does not exhibit a phase transition from a disordered state to an ordered state. The physical justification for this lies in the fact that forming a disordered phase from an orderedphase occurs through an energy change E = 4J in 1D (independent ofN ). In higher dimensions (D 2), the energy cost scales with N (e.g.E N 1/2J in 2D), thus leading to a finite-T phase transition (Fig. 3).Mean-field approximationThe solution for the Ising model for D

2 is not as easily found. In

Figure 3: Energy cost of a phase tran- sition from an ordered phase to a dis- ordered phase in 1D and 2D.1944, Lars Onsager found an exact solution for the 2D Ising model forh = 0 that exhibits a spontaneous magnetization for sufficiently largeJ . However, there is no exact solution for the Ising model for D 3.In this module we introduce the mean-field approximation, and the limits of this approximation. The mean-field approximation involves approximating the detailed interactions with the average of these inter- actions (Fig. 4).Define the average magnetization per site m = M

= 1 .N

(si) =mNi=1(s), noting that spatial invariance implies (si) = (s). Define the coor-dination number z as the number of nearest neighbors to any given site divided by 2 (e.g. z = 2 for 2D square lattice and z = 3 for 3D square lattice). The mean-field approximation replaces the near-neighbor in-teractions by the average interaction, thus sj (sj), giving the energyapproximation:

Figure 4: Schematic of the mean-field approximation.NNNE = h . si J . sisj h . si Jzm . si(8)i=1

(ij)

i=1

i=1which reduces the problem to an effective non-interacting model.Therefore, we write the average magnetization per site as:.m =s=1,1 s exp [(h + Jzm)s]s=1,1 exp [(h + Jzm)s]

= tanh(h + Jzm)(9)which performs the average of the site magnetization using the mean- field site energy in the Boltzmann weight for the spin probability.For the case h = 0 (no external field), the magnetization satisfiesm = tanh(Km), where K = Jz. This is a self-consistent mean-fieldequation. Local order is dictated by the surrounding order that is itself defined by the local order, leading to a self-consistent condition that must be met. Two scenarios exist (Fig. 5): For K < 1, there is only one solution for m that corresponds tom = 0. For K > 1, there are 3 solutions for m corresponding to m = 0 andm = m. The solution m = 0 is unphysical as it is a free energymaximum.Therefore, K = Kc = 1 defines a critical temperature Tc = Jz/kB , and we can write K = Tc/T (Fig. 6).Near T = Tc, we can expand the self-consistent equation near m = 0 to get:

Figure 5: Plots of tanh(Km) versus m for various values of K. Circles mark points that satisfy the self-consistent equation m = tanh(Km).

Figure 6: Phase diagram for the mean- field solution for the Ising model, show- ing m versus T /Tc.m = tanh

. Tc.m

Tc1 . Tc .3m

. 3m3 m

(Tc T )1/2TT3TTcGenerally, the external field h acts as the intensive variable that is conjugate to the magnetization M (extensive variable). In Module 2 we learned that thermodynamic stability dictates that a derivative of anintensive variable with respect to its conjugate extensive variable is a positive quantity, e.g.:. T . SHAPE \* MERGEFORMAT

S

V ,N

. p .,V

T ,N

, . i .Ni

T ,V ,Nj=i

> 0(10)Similarly, thermodynamic stability dictates that:. h . SHAPE \* MERGEFORMAT

MJz,N

> 0(11)The phase diagram for the mean-field solution is dictated by the self-consistent mean field equation (Fig. 7):m = tanh(h + Jzm)(12)This equation of state can be likened to the vapor-liquid equation of state p = p(v) (e.g. the van der Waals equation of state from Module 2, Fig. 9).The phase diagram for the mean-field approximation for the Isingmodel exhibits similar characteristics as the liquid-vapor phase diagram. Correspondence between M h in the magnetic system and p V inthe liquid-vapor system reflects the universal physical issues that are addressed with the Ising model and why it can be used to understand a variety of phenomena (Fig. 9).Critical point in vapor-liquid and magnetic systemsPreviously, we discussed the phase behavior of a magnetic system by analyzing the exact 1D and mean-field approximation of the Ising model. We saw that no phase transition occurs for the 1D Ising model and themodel exhibits a true phase transition for D 2, which was argued byphysical arguments and through the mean-field approximation.The h M phase diagram in a magnetic system was found to behave akin to the p V phase diagram in vapor-liquid systems. In both sys-tems, a critical point exists that marks the onset of phase coexistence in the system, i.e. for T > Tc, the system does not exhibit a phase transition, and for T < Tc, phase transitions exist. Now, we discuss the physical behavior near the critical point for both a magnetic system and a liquid-vapor system.The fluctuations of the energy, density, and magnetization for a sys- tem are found using various ensemble manipulations. From the canonical ensemble, the variance of the energy is:

Figure 7: Plot of m versus h for sev- eral values of K = Jz = Tc = T . The dashed part of each curve indicates where the system is unstable.

Figure 8: Comparison of the phase di- agrams for a magnetic system modeled using the mean-field approximation of the Ising model (left panel) and the vapor-liquid coexistence for a van der Waals fluid (right panel).

Figure 9: Plot of m versus h for K = Jz = Tc /T = 1.5. The arrows indi- cate the hysteresis in magnetization.(E ) (E) =

1 . 2Q . .

1 . Q .2 .Q2

..V ,N

Q2

..V ,N. 2 ln Q . .=

. 2F . .=

= kT 2CV(13)2

..V ,N

2

..V ,NThus connecting the energy fluctuations to CV . The standard devi-ation E = ,(E2) (E)

for energy fluctuations results in the charac-teristic magnitude of the energy fluctuation per molecule in terms of the heat capcity per molecule cV = CV as:

21=NkT 2c =

kT cV

(14)NNVNThis calculation verifies that the average energy per molecule is a well defined quantity with negligible fluctuations, provided that the heat capacity does not diverge. Similarly, the grand canonical ensemble for a one-component system yields the molecule number variance to be:(N ) (N)

. 2 ln . .=

. 2(pV ) . .=

= kT

. (N) . .()2

..,V

()2

..,V

.T ,V(15)And noting the Gibbs-Duhem equation for a 1 component system (d =sdT + vdp) and v = V , the variance in N can be written as:(N).

2

(N 2) (N)

V= kT v3

v.p.T

(16)Using the definition of molar density = 1 = (N)vV.

2

(2) ()

kT= V v3

v.p.T

(17).The condition for thermodynamic stability dictates that . v . . >p .T0, and if V , the density fluctuations in a stable system are neg-.

Figure 10: Plot of p versus v for a =3.375 = 27/8 (critical point, blue), a= 3.527, a = 3.679, a = 3.831, and a= 3.982 (red). The solid black curvesligible. However, the condition . p . .

0 dictates that the density

are the binodal curves (coexistence),v .Tfluctuations in the system diverge. Thus, the limit of stability for a

and the dashed black curves are the spinodal curves (limit of metastability).1-component system, marked by . p . .

= 0, exhibits wild density

The critical point is identified as thefluctuations.

v .T

point where the spinodal curves con- verge.As a system approaches the limit of stability for T < Tc, density fluctuations become substantial, leading to the eventual formation of the2 coexisting phases (spinodal decomposition) as can be seen in Fig. 10. As the system approaches the critical point, the density fluctuations dominate the physical behavior.Rayleigh Scattering occurs when the length scale of density fluctua- tions is comparable to the wavelength of light, where the local mismatch in the index of refraction leads to light scattering, as can be seen in Fig. 11.The fluctuation in the magnetization for the Ising model is governedby:

Figure 11: The sky is blue due to a phenomenon called Rayleigh scat- ter- ing, where light is scattered due to local density fluctuations in the fluid. Such density fluctuations occur for a fluid that is near its critical point, which is the case for the oxygen and nitrogen(M 2) (M)

. 2Q . .=

= . (M) . .

. m . .= N

in the upper atmosphere. The short- est wavelength light (violet end of spec-(h)2

..J ,N

(h)

..J ,N

(h)

..J(18)

trum) is scattered the most, leading to an overall perceived color that is light blue.As in the vapor-liquid system, the Ising model exhibits wild fluctua-tions at the limit of metastability, marked by the condition . (h) . .=M..J ,N0. In this case, the fluctuations are in the magnetization rather than thedensity. If we are far away from the critical point, the standard deviation in the magnetization per site is:.M =

1 . m . .

(19)NN(h)

...JThus, suggesting that the magnetization is well-defined in the ther- modynamic limit N . As we approach the critical point, the role offluctuations dominates the system thermodynamics, as we will proceed to discuss.Critical behavior for the Ising modelWe will discuss the critical behavior for the Ising model for the Mean- field approximation using our treatment developed earlier.To illustrate these concepts, focus on the 2-dimensional Ising modelon a square lattice. The analytical solution for the 2D Ising model shows that the critical temperature is Tc = 2.269J /k and the magnetization1scales as M N (T Tc) 8 .Mean-field approximationWe showed that the equation of state for the mean-field approximation to the Ising model is:m = tanh(h + Jzm) = tanh(h + Km)(20)The mean-field critical point is at h = 0 and Kc = 1, thus we canwrite K = Tc/T , where Tc = 2J /k for a 2D square lattice. Just belowT = Tc and h = 0, on expanding m near m = 0, we get:m = tanh

. Tc.m

Tc1 . Tc .3m

. 3m3 m

(Tc T )1/2TT3T

Tc(21)To find the fluctuations in the magnetization near T = Tc, we find:m2mh

= [1 tanh (h + Km)][1 + K h ]1 m2= 1 K + Km2

(22)For T > Tc, we have m = 0. Therefore:m11h

= 1 K T TAnd for T < Tc, we have:m1 3 + 3/K11h

= 1 K + K(3 3/K) K 1 T TThus, the mean-field approximation to the Ising model predicts the fluctuations in the magnetization to diverge near the critical temperature as:(M 2) (M)

N

|T Tc|

(23)For T = Tc (K = 1) and h = 0, expand m about m = 0 to get:m c (h + m) 1/3(h + m)3 m c 31/3h 1/3 h h1/3(24)Deviation of the mean-field approximation from the exact result lies in the neglect of correlations near the critical point, thus the mean-field approximation is best far from the critical point.For a vapor-liquid system governed by a mean-field theory (e.g. van der Waals equation of state), the critical behavior is:c (Tc T )1/2, (2) ( 2

1|T Tc|

T c (p pc)1/3(25)Which shows a direct correspondence between m and h p. Thecritical behavior of the Ising model and of other related thermodynamic systems reflects the underlying impact of correlations on the thermo- dynamic behavior. The mean-field approximation does not properly account for these correlations, a technique called renormalization group theory can account for spatial correlations which is beyond the scope of this course.si

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