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## II. DYNAMICS OF THE PFC MODEL WITH INERTIAL EFFECTS 

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 

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 

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and for the momentum density ⃗ g ( x, t ) 

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where η is a dissipative coefficient and the noise ⃗ ν ( ⃗ x, t ) has variance 

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The Poisson brackets for the mass and momentum den- sities are given by 16,17 

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Substituting Eqs. (6) into Eqs. (3) and (4) gives 

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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 

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To leading order in the momentum density ⃗ g , we drop the quadratic term in Eq. (8) giving 

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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 $^{21}$. 

## III. PFC MODEL WITH PINNING AND THERMAL FLUCTUATIONS 

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 

<img src='content_image/412779.jpg'>

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 

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