# Difference between dynamics of system evolving by Langevin equation with and without inertia

Each component of a many-particle system can be described by a Langevin equation:

$$m_i \frac{d^2r_i}{dt^2} = F_{config} + \gamma \frac{dr_i}{dt} + F_{noise}$$

Where $F_{config}$ depends on the configuration of the system (e.g. from a Lennard-Jones style potential) and $F_{noise}$ is the stochastic noise term. My question is, what difference does neglecting the acceleration term have on the dynamics of the system? I can imagine that it would not affect the equilibrium configuration. What about non-equilibrium steady states?

Non-dimensionalizing the distance and time using $r_i^* = \frac{r_i}{\sigma}$ and $\tau = \frac{t}{T_c}$ where $T_c$ is some "typical time" to be specified.

The equation becomes:

$$\frac{m_i\sigma}{T_c^2} \frac{d^2r^*_i}{d\tau^2} = F_{config} + \frac{\gamma \sigma}{T_c} \frac{dr_i^*}{d\tau} + F_{noise}$$

It can be seen that if $T_c >> \frac{m}{\gamma}$, the acceleration term can be safely neglected. Does this mean that say if I run two simulations of the system, one using Langevin dynamics, the other using overdamped Langevin dynamics, and take snapshots of each simulation, if the snapshots are far enough apart in time, I will get the same results from both simulations?

• What does the stochastic term add here? Doing just the ODE, what would your conclusions be then? – alarge Apr 2 '18 at 10:21
• Just doing the ODE, I would conclude that at long enough time scales, the behaviour would be the same. – The Hagen Apr 2 '18 at 15:25
• That's wrong conclusion. Consider deterministic vs stochastic behavior of particle in harmonic trap; ensemble of the first oscillates in the phase space, just differ by phase from the initial condition, ensemble of the second reaches thermal steady state and forgets any initial condition - very different from the deterministic case. The only way to get the second result from simulations of the first are simulations inaccuracies which present randomization of the dynamics. However even if it reaches the same steady state - it reaches it very differently and by completely different timescale – Alexander Apr 5 '18 at 22:20
• @Alexander, either I misunderstood alarge or you misunderstood me. My original question was, if you compare the dynamics of an overdamped Langevin system with a normal Langevin system at long timescales, if there would be a difference. In both cases, the behaviour is stochastic and not deterministic. – The Hagen Apr 6 '18 at 15:54
• My previous comment was, that if you neglect the random term in both Langevin equations (overdamped and not), I would expect this to be true from dimensional analysis. – The Hagen Apr 6 '18 at 15:54