Unformatted text preview:

MIT OpenCourseWare http://ocw.mit.edu 8.334 Statistical Mechanics II: Statistical Physics of Fields Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.I.C Phase Transitions The most spectacular consequence of interactions among particles is t he a ppearance of new phases of matter whose collective behavior bears little resemblance to that of a few particles. How do the particles then transform from one macroscopic stat e to a completely different one. From a formal perspective, all macroscopic properties can be deduced from the free energy or the partition function. Since phase transitions typically involve dramatic changes in various response functions they must correspond to singularities in the free energy. The canonical partition function for a finite collection of particles is always an analytical function. Hence phase transitions, and their associated non–analyticities, are only obtained for i nfinitely ma ny particles, i.e. in the thermodynamic limit, N → ∞. The study of phase t ransitions is thus rela ted to finding the origin o f various singularities in the free energy and characterizing them. The classical example of a phase transition is the condensation of a gas into a liquid. Some important features of the l iquid–gas condensation transition are: (1) In the temperature/pressure plane, (T, P ), the phase t ra nsition occurs along a line that terminates at a critical point (Tc, Pc). (2) In the volume/pressure plane, (P, v ≡ V/N) , the tra nsition appears as a coexis te nce interval, corresponding to a mixture of g as a nd liquids of densities ρg = 1/vg, and ρl = 1/vl, at temperatures T < Tc. (3) Due to the termination of the coexistence li ne, it is possible to go from the gas phase to the li quid phase continuously (w ithout a phase transition) by going around the critical point. Thus there are no fundamental differences between liquid and g as phases. From a mat hemati cal perspective, t he free energy of the sy stem is an analyt ical func-tion in the (P, T ) plane, except for some form of branch cut along t he phase boundary. Observations in the vicinity of the criti cal point further indicate that : (4) The difference between the densities of coexisting liq uid and gas phases vanishes on approaching Tc, i.e. ρliquid ρgas, as T Tc − .→ →(5) The pressure versus volume isotherms become progressively mo re flat on approaching TC from the high temperature side. This implies that the isothermal compressibility, κT = − ∂V/∂P|T /V , diverges as T → Tc + . (6) The fluid appears “milky” close to crit icality. This phenomenon, known as critical opalescence, suggests collective fluctuations in the gas at long enough wavelengths to scatter visible light. These fluctuations must necessarily involve many particl es, and a coarse graining procedure may thus be a ppropriate to t heir description. 10� A related, but possibly less familia r, phase transition occurs between paramagnetic and ferromagnetic phases of certain substances such as iron or nickel. These materials become spontaneously magnetized below a Curie temperature Tc. There is a discontinuity in mag netizat ion of the substance as the magnetic field h, goes through zero for T < Tc. The phase diagram in the (h, T ) plane, and the magnetization isotherms M( h), have much in common with their counterparts in the condensation problem. In both cases a line of discontinuous transitions terminates at a critical point, and the isotherms exhibit singular behavior in the vici nity of this point. The phase diagram of the magnet is simpler in appearance, because the symmetry h 7→ −h ensures that the critical point occurs a t hc = Mc = 0. I.D Critical Behavior The singular behavior in the vicinity of a critical po int is characterized by a set of critical exponents. These exponents describe the non–analyticity of vario us thermodynamic functions. The most commonly encountered exponents are li sted below: • The Order Parameter: By definition, there is more than one equilibrium phase on a coexistence line. The order parameter is a thermodynamic function that is different in each phase, and hence can be used to distinguish between them. For a magnet, the magnetization 1 m(T ) = lim M(h, T ),V h→0 serves as the order parameter. In zero field, m vanishes for a paramagnet and is non–zero in a ferromagnetic, i.e. 0 for T > Tc, m(T, h = 0) ∝|t|β for T < Tc, (I.20) where t = (Tc − T )/Tc is the reduced temperature. The singular behavior of the order parameter along the coexistence line is therefore indicated by the exponent β. The singular behavior of m along the critical i sot herm is indicated by another exponent δ, defined through m(T = Tc, h) ∝ h1/δ. (I.21) The two phases along the liq uid–gas coexistence line are differentiated by their density, and the density difference ρ − ρc, where ρc is the critical density, serves the role of the order parameter. 11� � � � � � � • Response Functions: The critical system is quit e sensitive to ext ernal perturbations, as typified by the infinite compressibility at the liquid–gas critical point. The divergence in t he response of the order parameter t o a field conjugate to it is indicated by an exponent γ. For example, in a magnet, χ±(T, h = 0) ∝ | t|−γ± , (I.22) where in principle two exponents γ+ and γ− are necessary to describe the divergences on the two sides of the phase transition. Actually in almost all cases, the same singularity governs both sides and γ+ = γ− = γ. The heat capacity is the thermal response function, and its singularities at zero field are described by the exponent α, i.e. C±(T, h = 0 ) ∝ |t|−α± . (I.23) • Long–range Correlations: Since the response functions


View Full Document
Download I.C Phase Transitions
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view I.C Phase Transitions and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view I.C Phase Transitions 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?