Looking at your derivation, I can see several issues with the proposed constant action term from collisions:

Conceptual Issues

1. Action principles don't include collisions directly: In classical mechanics, the action integral is defined for smooth trajectories. Collisions represent discontinuities where the standard variational principle breaks down. The action formulation assumes differentiable paths, but collisions involve impulses (discontinuous momentum changes).

2. Double counting: If you're using a potential U that is zero except during collisions, then the collision effects are already captured in the -U term of your action. Adding an additional collision term would be double counting.

3. Dimensional analysis problems: Your expression has several dimensional inconsistencies:

   • has dimensions of [Force] if V is a potential
   • appears to be a chemical potential with dimensions [Energy]
   • seems to be a mean free path with dimensions [Length]
   • The combination has dimensions [Energy × Length / Energy × Temperature⁻¹] = [Length × Temperature]

   This doesn't give the correct dimensions for action [Energy × Time].

Technical Issues

4. Constant terms in action: While it's true that constant terms don't affect equations of motion in classical mechanics, your term isn't actually constant - it depends on N, R, and other parameters that could vary.

5. Statistical mechanics confusion: You seem to be mixing single-particle mechanics (action principle) with statistical mechanics concepts (chemical potential, mean free path). The action principle applies to individual particle trajectories, not ensemble averages.

Alternative Approaches

Instead of modifying the action, consider:

• Kinetic theory: Use the Boltzmann equation to properly account for collisions statistically
• Hard sphere potentials: Use proper collision potentials in your original action
• Effective field theory: If you want coarse-grained dynamics, derive effective equations that average over collision timescales

The standard approach in molecular dynamics is to either:

1. Use continuous potentials (like Lennard-Jones) that naturally handle "collisions"
2. Use event-driven dynamics for hard spheres, where collisions are handled as boundary conditions between smooth trajectories

Your equation (2) appears to be an ad hoc construction that doesn't follow from fundamental principles of mechanics or statistical physics.