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Formulation of MZ-guided Markovian/Non-Markovian DPD
The equation of motion (EOM) of the coarse-grained (CG) particles obtained from the Mori-Zwanzig projection is given by
- Equation 1:
ddtPI=1β∂∂RIlnω(R)−βK∑X=1∫t0ds⟨[δFI(t−s)][δFX(0)]T⟩PX(s)MX+δFI(t) ,
where β=1/kBT with T the thermodynamic temperature and kB the Boltzmann constant, R={R1,R2,⋯,RK} is a phase point in the CG phase space, and ω(R) is defined as a normalized partition function of all the microscopic configurations at phase point R given by
- Equation 2:
ω(R)=∫dNˆrδ(ˆR−R)e−βU∫dNˆre−βU ,
where U is the potential energy corresponding to the phase point R, and the integrations are performed over all the possible microscopic configurations {ˆri}.
In the right-hand side of Eq.(???), the first term represents the conservative force due to the change of microscopic configuration, and it is the ensemble average force on cluster I denoted as ⟨FI⟩. The last term δFI is the fluctuating force on cluster I and it is given by δFI=FI−⟨FI⟩ in which FI is the instantaneous total force acting on the cluster I. The second term of Eq. (???) is the friction force determined by an integral of memory kernel.
First approximation: Here, we assume that the non-bonded interactions between neighboring clusters in the microscopic system are explicitly pairwise decomposable, and hence the total force consists of pairwise forces, e.g. FI≈∑J≠IFIJ and δFI≈∑J≠IδFIJ.
However, when we consider the force FIJ that a molecule J exerts on another molecule I, in principle, FIJ involving many-body effects depends on all the COM coordinates R as well as their microscopic configurations. Although Eq. (???) based on the Mori-Zwanzig formalism is accurate, a direct computation of the many-body interactions is very difficult.
Second approximation: In practice, we neglect the many-body correlations between different pairs, and assume that the force FIJ between two clusters I and J depends only on the relative COM positions RI and RJ and is independent of the positions of the rest of clusters.
Therefore, the conservative term is in Eq.(???) approximated by 1β∂∂RIlnω(R)=⟨FI⟩≈∑J≠I⟨FIJ⟩=∑J≠IFCIJ(RIJ)eIJ
where eIJ is the unit vector from CG particle J to I given by eIJ=(RI−RJ)/RIJ with RIJ=|RI−RJ|. The rotational symmetry of the CG pairs about the eIJ suggests that, on average, FIJ has no preference on the plane perpendicular to eIJ and remains only nonzero component along eIJ. Here, FCIJ(RIJ) represents the magnitude of conservative force FCIJ, which is time independent but distance dependent.
Similarly, the fluctuating force defined as the deviation from the mean force is also decomposed into pairwise forces δFI=FI−⟨FI⟩≈∑J≠IδFIJδFIJ=FIJ−⟨FIJ⟩=FIJ−FCIJ(RIJ)eIJ
where FIJ is the instantaneous force exerted by cluster J on cluster I, and ⟨FIJ⟩ is the ensemble average of FIJ obtained by Eq. (???).
Now we consider the memory kernel in Eq. (???). Based on the second approximation, the correlation of fluctuating forces between different pairs is ignored. Thus, we have ∑J≠I∑Y≠X⟨[δFIJ(t−s)][δFXY(0)]T⟩VX(s)=⟨[δFIJ(t−s)][δFIJ(0)]T⟩VI(s)|X=I,Y=J+⟨[δFIJ(t−s)][δFJI(0)]T⟩VJ(s)|X=J,Y=I=⟨[δFIJ(t−s)][δFIJ(0)]T⟩VIJ(s)
where VIJ=VJ−VJ is the relative velocity of CG particle I to J. Moreover, {\color{red}Third approximation:} we assume that the memory on time is finite, e.g. history length N⋅Δt where Δt is the time step of DPD simulations. Therefore, the time correlation between the fluctuating forces is zero when the time interval is larger than NΔt ⟨[δFI(t)][δFX(0)]T⟩|t>NΔt=0
Then, the second term in Eq.(???) can be expanded as follows:
−βK∑X=1∫t0ds⟨[δFI(t−s)][δFX(0)]T⟩VX(s)=−βK∑X=1∫tt−NΔtds⟨[δFI(t−s)][δFX(0)]T⟩VX(s)=−β⋅ΔtK∑X=1N∑n=0αn⟨[δFI(nΔt)][δFX(0)]T⟩VX(t−nΔt)=−β⋅ΔtK∑X=1∑J≠I∑Y≠XN∑n=0αn⟨[δFIJ(nΔt)][δFXY(0)]T⟩VX(t−nΔt)=−β⋅Δt∑J≠IN∑n=0αn⟨[δFIJ(nΔt)][δFIJ(0)]T⟩VIJ(t−nΔt)
We define the friction matrix ΓIJ,n as ΓIJ,n=β⟨[δFIJ(nΔt)][δFIJ(0)]T⟩
where δFIJ is the fluctuating force. Generally, the fluctuating force δFIJ is not parallel to the radial direction eIJ. However, δFIJ, on average, is transversely isotropic with respect to eIJ because the instantaneous pairwise force FIJ has no preference between directions ⊥1 and ⊥2.
When we calculate the friction matrix, we do not distinguish between the directions ⊥1 and ⊥2 and decompose δFIJ into two parts
δFIJ=(eIJeTIJ)⋅δFIJ+(1−eIJeTIJ)⋅δFIJ=δF∥IJ+δF⊥IJ ,
where δF∥IJ is the component along vector eIJ and δF⊥IJ the perpendicular part whose modulus is equally distributed on directions ⊥1 and ⊥2.
Remark: The memory term given by Eq. (???) can be further simplified with a {\color{red}Markovian assumption} that the memory of fluctuating force in time is short enough to be approximated by a Dirac delta function
β⟨[δFIJ(t−s)][δFIJ(0)]T⟩=2γIJδ(t−s) ,β∫t0ds⟨[δFIJ(t−s)][δFIJ(0)]T⟩VIJ(s)=γIJ⋅VIJ(t) ,
where γIJ is the friction tensor defined by γIJ=β∫∞0dt⟨[δFIJ(t)][δFIJ(0)]T⟩. Then, the equation of motion of DPD particles based on the Markovian approximation can be expressed by
dPIdt=∑J≠I{FCIJ(RIJ)eIJ−γIJ(RIJ)(eIJ⋅VIJ)eIJ+δFIJ(t)}