How Do DOLLS and SLLOD Algorithms Differ in Molecular Dynamics?

[Tilde90]

Apparently, this is the DOLLS tensor Hamiltonian:

[ tex ] H = H_0 + \sum q_i p_i : ∇u(t)^T [ /tex ]

These are the derived equations of motion:

[ tex ] \dot{q}_i = p_i/m + q_i \cdot ∇u [ /tex ]

[ tex ] \dot{p}_i = F_i - ∇u \cdot p_i [ /tex ]

And these are the SLLOD equations of motion:

[ tex ] \dot{q}_i = p_i/m + q_i \cdot ∇u [ /tex ]

[ tex ] \dot{p}_i = F_i - p_i \cdot ∇u [ /tex ]

For me this is nonsense. What is a division of an outer product (I guess), [ tex ] \sum q_i p_i [ /tex ], by a transposed vector? And, most of all, what is the difference between the equations of motion of these two methods?

 

[eljose79]

Thank you for bringing up this topic. I can understand your confusion and concern regarding the DOLLS tensor Hamiltonian and its derived equations of motion.

Firstly, let me clarify that the DOLLS tensor Hamiltonian is a mathematical representation of the dynamics of a system, specifically in the context of molecular dynamics simulations. It is derived from the Lagrangian formalism and is commonly used in studies of molecular systems.

Now, to address your questions, let's break down the equations and understand them step by step.

Starting with the DOLLS tensor Hamiltonian, [ tex ] H = H_0 + \sum q_i p_i : ∇u(t)^T [ /tex ], the notation [ tex ] \sum q_i p_i [ /tex ] represents the sum of the outer product of the position and momentum vectors of all the particles in the system. This is a common notation used in molecular dynamics simulations to represent the total kinetic energy of the system.

The division by the transposed vector [ tex ] ∇u(t)^T [ /tex ] is a result of the Hamiltonian formalism and is used to transform the equations of motion from the Lagrangian to the Hamiltonian formalism. This transformation is necessary for numerical simulations and allows for more efficient calculations.

Moving on to the derived equations of motion, [ tex ] \dot{q}_i = p_i/m + q_i \cdot ∇u [ /tex ] and [ tex ] \dot{p}_i = F_i - ∇u \cdot p_i [ /tex ], these are simply the time derivatives of the position and momentum vectors, respectively. The term [ tex ] q_i \cdot ∇u [ /tex ] represents the force experienced by the particle due to the potential energy gradient, while [ tex ] F_i [ /tex ] represents the total force acting on the particle.

Finally, the SLLOD equations of motion, [ tex ] \dot{q}_i = p_i/m + q_i \cdot ∇u [ /tex ] and [ tex ] \dot{p}_i = F_i - p_i \cdot ∇u [ /tex ], are derived from the DOLLS tensor Hamiltonian using the Symplectic Low-Order Lie Derivative (SLLOD) method. This method is used to improve the accuracy and stability of numerical simulations.

In summary, the DOL