2 II. DYNAMICS OF THE PFC MODEL WITH INERTIAL EFFECTS III. PFC MODEL WITH PINNING AND THERMAL FLUCTUATIONS In the generalized PFC model, the system is repre- sented by a coarse grained effective Hamiltonian that is a functional of the number density field. To take into account inertial effects in the dynamics, we need to con- sider the contribution of the kinetic energy to the to- tal energy of the system in addition to the configura- tional energy. Thus we consider the momentum den- sity, ⃗ g ( ⃗ x ) = ρ ( ⃗ x ) ⃗ v ( x ), a dynamical variable in the coarse grained Hamiltonian in addition to the particle density field ρ ( ⃗ x ). The total effective Hamiltonian in the pres- ence of an external force ⃗ f can be written as where H$_{kin}$ is the kinetic energy contribution given by Substituting Eqs. (6) into Eqs. (3) and (4) gives for a spatially uniform external force ( f$_{i}$ ). We can rede- fine the coefficient η → ρη to remove the denominator from the term g$_{i}$/ρ in Eq. (8). With this change the variance of the noise ⃗ ν ( x, t ) becomes To leading order in the momentum density ⃗ g , we drop the quadratic term in Eq. (8) giving and H$_{int}$ ( ρ ) is the configurational contribution to the ef- fective Hamiltonian in the original PFC model. The last term is due to the presence of the the external force ⃗ f . In the absence of energy dissipation, the time depen- dence of ρ and ⃗ g are determined by the Poisson brack- ets { H$_{t}$, ρ } and { H$_{t}$, ⃗ g } . At finite temperatures, addi- tional dissipative noise terms are present in the dynam- ical equations. The noise satisfies the fluctuation dis- sipation relations 16,17 which allows the system to reach thermal equilibrium in the absence of external perturba- tions. We include the dissipative noise term directly in the dynamical equations for ⃗ g , which is a nonconserved field. For the particle density ρ ( ⃗ x, t ) we have Similar dynamical equations for PFC models with inter- nal dissipation were obtained in Ref. 20. If the effective Hamiltonian H$_{int}$ is known, then these coupled stochas- tic dynamical equations should provide a full description of the particle and momentum densities in presence of fluctuations represented by the noise ν$_{i}$ ( x, t ) with corre- lations proportional to the temperature T and inertial effects determined by the damping parameter η . In the overdamped limit, ∂g/∂t = 0, with T = 0 and f = 0, the equation for the time evolution of ρ obtained by inserting ⃗ g from Eq. (11) in Eq. (10) reduces to the deterministic equation for the density which has been obtained from classical density functional theory of liquids $^{21}$. In the presence of an external pinning potential $^{13–15}$, a specific form of the configurational energy contribu- tion H$_{int}$ to the total effective Hamiltonian in Eq.(1) has been proposed which is an extension of the stan- dard PFC model free energy functional used in many applications $^{11,12}$. In dimensionless form, this effective interaction Hamiltonian H$_{int}$ = H$_{pfc}$ can be written as where η is a dissipative coefficient and the noise ⃗ ν ( ⃗ x, t ) has variance and for the momentum density ⃗ g ( x, t ) The Poisson brackets for the mass and momentum den- sities are given by 16,17 where ψ ( ⃗ x ) is a continuous field, V ( ⃗ x ) represents the ex- ternal pinning potential and r is a parameter. The phase field ψ ( ⃗ x ) can be regarded as a measure of deviations of H$_{pfc}$ = ∫ d⃗ x { 1 2 ψ [ r + (1 + ∇ $^{2}$)$^{2}$] ψ + ψ 4 4 + V ψ } , (12) ∂g$_{i}$ ∂t = − ρ ∇$_{i}$ δH$_{int}$ δρ + ρf$_{i}$ − ηg$_{i}$ + ν$_{i}$ ( ⃗ x, t ) . (11) ∂ρ ∂t = −∇ · ⃗ g ; (10) 〈 ν$_{i}$ ( ⃗ x, t ) ν$_{j}$ ( ⃗ x $^{′}$, t$^{′}$) 〉 = 2 k$_{B}$T ηρ ( ⃗ x ) δ ( ⃗ x − ⃗ x $^{′}$) δ ( t − t $^{′}$) δ$_{i,j}$ (9) ∂g$_{i}$ ∂t = − ρ ∇$_{i}$ δH$_{int}$ δρ − η g$_{i}$ ρ + ρf$_{i}$ + ν$_{i}$ ( ⃗ x, t ) (8) − ∑ j ∇$_{j}$ g$_{i}$g$_{j}$ ρ , ∂ρ ∂t = −∇ · ⃗ g ; (7) H$_{t}$ = H$_{kin}$ + H$_{int}$ − ∫ d⃗ xρ ( ⃗ x ) ⃗ x · ⃗ f , (1) H$_{kin}$ = ∫ d⃗ x ⃗ g $^{2}$( ⃗ x ) 2 ρ ( ⃗ x ) , (2) ∂ρ ∂t = − ∑ j ∫ d⃗ x $^{′}${ ρ ( ⃗ x ) , g$_{j}$ ( ⃗ x $^{′}$) } δH$_{t}$ δg$_{j}$ ( ⃗ x ′ ) , (3) ∂g$_{i}$ ∂t = − ∫ d⃗ x $^{′}${ g$_{i}$ ( ⃗ x ) , ρ ( ⃗ x $^{′}$) } δH$_{t}$ δρ ( ⃗ x ′ ) − ∑ j ∫ d⃗ x $^{′}${ g$_{i}$ ( ⃗ x ) , g$_{j}$ ( ⃗ x $^{′}$) } δH$_{t}$ δg$_{j}$ ( ⃗ x ′ ) − η δH$_{t}$ δg$_{i}$ + ν$_{i}$ ( ⃗ x, t ) , (4) 〈 ν$_{i}$ ( ⃗ x, t ) ν$_{j}$ ( ⃗ x $^{′}$, t$^{′}$) 〉 = 2 k$_{B}$T ηδ ( ⃗ x − ⃗ x $^{′}$) δ ( t − t $^{′}$) δ$_{i,j}$ . (5) { ρ ( ⃗ x ) , g$_{i}$ ( ⃗ x $^{′}$) } = ∇$_{i}$ ( ρ ( ⃗ x ) δ ( ⃗ x − ⃗ x $^{′}$)); (6) { g$_{i}$ ( ⃗ x ) , ρ ( ⃗ x $^{′}$) } = ρ ( ⃗ x ) ∇$_{i}$ δ ( ⃗ x − ⃗ x $^{′}$); { g$_{i}$ ( ⃗ x ) , g$_{j}$ ( ⃗ x $^{′}$) } = ∇$_{j}$ ( g$_{i}$ ( ⃗ x ) δ ( ⃗ x − ⃗ x $^{′}$)) − ∇ ′ $_{i}$( g$_{j}$ ( ⃗ x ) δ ( ⃗ x − ⃗ x $^{′}$)) .