Past exam of the thermal and statistical physics course of the University of Cambridge / 2012 / Paper 1 / 2

Landau theory expands a thermodynamic potential in an order parameter, constrained by symmetry. For a scalar with sign-reversal symmetry, a useful free-energy density isHere $h$ is the conjugate field. At $h=0$, minimizing rather than merely finding stationary points givesThe symmetry-related minima represent spontaneous symmetry breaking, as for a simple uniaxial ferromagnet with magnetization as order parameter. An infinitesimal field selects one sign. Above $T_c$, the susceptibility is $1/[a(T-T_c)]$; below it the stable-branch susceptibility is $1/[2a(T_c-T)]$. Minimization lowers $f$ by $-a^2(T-T_c{)}^2/(4b)$ below $T_c$, producing a heat-capacity density jump $T_c{a}^2/(2b)$ with continuous entropy.

A negative quartic coefficient requires a stabilizing $+c{\eta }^6/6$ with $c>0$. It can produce a first-order transition before the quadratic coefficient changes sign: coexistence occurs at $a(T-T_c)=3b^2/(16c)$ and ${\eta }^2=-3b/(4c)$. Odd terms are allowed when the assumed symmetry is absent. Landau theory predicts mean-field exponents; spatial fluctuations can change critical behaviour, especially close to a critical point in low dimensions. A gradient term extends the model to nonuniform order parameters but does not by itself supply dynamical equations.

Brownian motion arises from many microscopic collisions. A one-dimensional Langevin description isThe fluctuation-dissipation relation ties random forcing to drag, ensuring thermal equilibrium. For a free particle set $\kappa =0$. With initially thermal velocities, its mean-square displacement isIt is ballistic, $(k_{B}T/m)t^2$, at short times and diffusive, $2Dt$, for $t\gg {\tau }_v$. In three dimensions the corresponding long-time result is $6Dt$. At diffusive times, the position propagator from a specified origin is Gaussian with variance $2Dt$ per coordinate. A small sphere in a viscous liquid supplies a physical example; in the slow unbounded-fluid limit its drag is $6\pi {\eta }_{f}a$.

For a harmonic trap, overdamped relaxation has time ${\tau }_{\kappa }=\zeta /\kappa$, equilibrium variance $k_{B}T/\kappa$ and stationary correlationConsequently stationary trapped Brownian displacement obeysConfinement makes this displacement saturate, unlike free diffusion. A trajectory initialized exactly at the centre instead has variance $(k_{B}T/\kappa )(1-e^{-2t/{\tau }_{\kappa }})$; confusing that conditional variance with the stationary displacement gives an erroneous factor of two. The equilibrium position distribution is proportional to $\exp [-\kappa x^2/(2k_{B}T)]$. More general finite confinement has a geometry-dependent equilibrium distribution and bounded displacement, not necessarily a Gaussian one.

Bose-Einstein condensation is macroscopic occupation of a single lowest-energy one-particle state. For a homogeneous three-dimensional ideal gas of conserved massive bosons, set that energy to zero. The excited-state population isHere $g_{3/2}(z)={\sum }_{r\ge 1}{z}^r/r^{3/2}$. As $\mu \to 0^{-}$, the excited states can hold only $V\zeta (3/2)/{\lambda }_T^3$ particles. Additional particles occupy the ground state. ThusThe isolated ground state must be counted separately from the continuum density of states; otherwise one misses the condensate. A dilute atomic gas is a physical realization of the quantum-degeneracy regime, but trapping geometry and interactions modify the homogeneous ideal-gas formulas.^[ref]

In the homogeneous ideal gas, one and two dimensions do not have finite-temperature condensation in the thermodynamic limit because the low-energy population integral diverges. Finite size or a trap changes that conclusion's assumptions. Condensation and superfluidity are distinct: the ideal quadratic gas has zero Landau critical velocity, while interactions can support a sound mode and superfluid flow. Neither zero viscosity nor phase-rigid flow follows from the occupation formula alone.