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Abstract / Description of output
We introduce a versatile bottomup derivation of a formal theoretical framework to describe (passive) softmatter systems out of equilibrium subject to fluctuations. We provide a unique connection between the constituentparticle dynamics of real systems and the time evolution equation of their measurable
(coarsegrained) quantities, such as local density and velocity. The starting point is the full Hamiltonian description of a system of colloidal particles immersed in a fluid of identical bath particles. Then, we average out the bath via Zwanzig’s projectionoperator techniques and obtain the stochastic Langevin equations governing the colloidalparticle dynamics. Introducing the appropriate definition of the local number and momentum density fields yields a generalisation of the Dean–Kawasaki (DK)model, which resembles the stochastic Navier–Stokes description of a fluid. Nevertheless, the DK equation still contains all the microscopic information and, for that reason, does not represent the dynamical law of observable quantities. We address this controversial feature of the DK description by carrying out a nonequilibrium ensemble average. Adopting a natural decomposition into localequilibrium and nonequilibrium contribution, where the former is related to a generalised version of the canonical distribution, we finally obtain the fluctuatinghydrodynamic equation governing the timeevolution of the mesoscopic density and momentum fields. Along the way, we outline the connection between the ad hoc energy functional introduced in previous DK derivations and the freeenergy functional from classical densityfunctional theory. The resultant equation has the structure of a dynamical density functional theory (DDFT) with an additional fluctuating force coming from the random interactions with the bath. We show that our fluctuating DDFT formalism corresponds to a particular version of the fluctuating Navier–Stokes equations, originally derived by Landau and Lifshitz. Our framework thus provides the formal apparatus for ab initio derivations of fluctuating DDFT equations capable of
describing the dynamics of softmatter systems in and out of equilibrium.
(coarsegrained) quantities, such as local density and velocity. The starting point is the full Hamiltonian description of a system of colloidal particles immersed in a fluid of identical bath particles. Then, we average out the bath via Zwanzig’s projectionoperator techniques and obtain the stochastic Langevin equations governing the colloidalparticle dynamics. Introducing the appropriate definition of the local number and momentum density fields yields a generalisation of the Dean–Kawasaki (DK)model, which resembles the stochastic Navier–Stokes description of a fluid. Nevertheless, the DK equation still contains all the microscopic information and, for that reason, does not represent the dynamical law of observable quantities. We address this controversial feature of the DK description by carrying out a nonequilibrium ensemble average. Adopting a natural decomposition into localequilibrium and nonequilibrium contribution, where the former is related to a generalised version of the canonical distribution, we finally obtain the fluctuatinghydrodynamic equation governing the timeevolution of the mesoscopic density and momentum fields. Along the way, we outline the connection between the ad hoc energy functional introduced in previous DK derivations and the freeenergy functional from classical densityfunctional theory. The resultant equation has the structure of a dynamical density functional theory (DDFT) with an additional fluctuating force coming from the random interactions with the bath. We show that our fluctuating DDFT formalism corresponds to a particular version of the fluctuating Navier–Stokes equations, originally derived by Landau and Lifshitz. Our framework thus provides the formal apparatus for ab initio derivations of fluctuating DDFT equations capable of
describing the dynamics of softmatter systems in and out of equilibrium.
Original language  English 

Article number  123022 
Number of pages  17 
Journal  New Journal of Physics 
Volume  19 
DOIs  
Publication status  Published  7 Dec 2017 
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 1 Finished

Statistical mechanics of soft matter: Derivation, analysis and implementation of dynamic density functional theories
30/11/14 → 29/11/17
Project: Research