[Paper Review] A Time-Asymmetric Process in Central Force Scatterings

Ramis Movassagh|arXiv (Cornell University)|Aug 4, 2010

Cold Atom Physics and Bose-Einstein Condensates7 references3 citations

TL;DR

This paper identifies a time-asymmetric energy exchange mechanism in central force scattering, where a fast, light particle in an attractive force field (e.g., gravitational or Coulomb) statistically loses energy when scattering off slowly moving, massive particles due to asymmetric potential well dynamics during transverse collisions. The key result is a derived formula (Eq. 12) quantifying this net energy loss proportional to $ \frac{\alpha z_0 N}{j^2 v_2^2} \log\left(\frac{D_{\text{max}}}{D_{\text{min}}}\right) $, with opposite behavior in repulsive fields where energy gain occurs.

ABSTRACT

This article puts forth a process applicable to central force scatterings. Under certain assumptions, we show that in attractive force fields a high speed particle with a small mass speeding through space, statistically loses energy by colliding softly with large masses that move slowly and randomly. Furthermore, we show that the opposite holds in repulsive force fields: the small particle statistically gains energy. This effect is small and is mainly due to asymmetric energy exchange of the transverse (i.e., perpendicular) collisions. We derive a formula that quantifies this effect (Eq. 12). We then put this work in a broader statistical context and discuss its consistency with established results.

Motivation & Objective

• To investigate statistical energy changes in a fast, light particle undergoing repeated two-body scatterings in a dilute system of massive, randomly moving particles.
• To identify and quantify a net energy transfer effect arising from transverse collisions in central force fields.
• To establish the time-asymmetric nature of this energy exchange, breaking detailed balance in scattering processes.
• To derive a closed-form expression for the average energy change of the light particle under small-angle scattering assumptions.
• To place the result in the context of classical kinetic theory and statistical mechanics, ensuring consistency with established frameworks such as the Fokker-Planck approximation.

Proposed method

• Model a fast, light particle ($ m_2 $) scattering off a dilute ensemble of massive, slowly moving particles ($ m_1 $) in a central potential $ V(r) = \alpha / r^k $, with $ \alpha < 0 $ for attraction.
• Analyze the energy exchange during small-angle, transverse collisions by decomposing the scattering into near-collisional dynamics where the force varies with distance.
• Use perturbative expansion in $ v_1 / v_2 \ll 1 $ to compute the average energy change $ \langle \Delta E \rangle $, focusing on the leading-order asymmetry in potential well depth during approach vs. recession.
• Integrate over the velocity distribution of the massive particles using the Maxwellian-like form $ N(v_1) dv_1 = \frac{4j^3}{\sqrt{\pi}} N e^{-j^2 v_1^2} v_1^2 dv_1 $, typical in kinetic theory.
• Account for all impact parameters by integrating over $ D_0 $ with logarithmic divergence handled via physical cutoffs $ D_{\text{min}} $ and $ D_{\text{max}} $, consistent with astrophysical and plasma physics conventions.
• Derive the final expression $ \langle \Delta E \rangle_{D_0, \mathbf{v}_1} = \frac{2\alpha z_0 N}{j^2 v_2^2} \log\left(\frac{D_{\text{max}}}{D_{\text{min}}}\right) $, showing net energy loss in attractive fields.

Experimental results

Research questions

• RQ1 Does a net energy transfer occur in a sequence of two-body scatterings between a fast light particle and randomly moving massive particles in a central force field?
• RQ2 Is this energy transfer time-asymmetric, such that the statistical outcome differs between approach and recession of the massive particle?
• RQ3 What is the quantitative magnitude of the average energy change $ \langle \Delta E \rangle $ for the light particle under small-angle scattering?
• RQ4 How does the energy exchange depend on the relative velocity $ v_1 / v_2 $, impact parameter $ D_0 $, and the field particle density $ N $?
• RQ5 Does the result remain consistent with established kinetic theory and relaxation time approximations in astrophysics and plasma physics?

Key findings

• A net energy loss occurs for a fast, light particle in an attractive central force field due to time-asymmetric potential well dynamics during transverse collisions.
• The average energy change is quantified by $ \langle \Delta E \rangle_{D_0, \mathbf{v}_1} = \frac{2\alpha z_0 N}{j^2 v_2^2} \log\left(\frac{D_{\text{max}}}{D_{\text{min}}}\right) $, with $ \alpha < 0 $ implying net energy loss.
• In repulsive fields ($ \alpha > 0 $), the same mechanism leads to net energy gain for the light particle.
• The effect arises primarily from transverse collisions, not head-on ones, and is proportional to $ (v_1 / v_2)^2 $, with higher-order corrections negligible under small-angle scattering.
• The result is consistent with classical kinetic theory and relaxation time models, particularly in the context of stellar dynamics and plasma physics.
• The logarithmic divergence in $ D_0 $ is physically regularized by $ D_{\text{min}} $ (for small-angle validity) and $ D_{\text{max}} $ (for system density), with the result robust to moderate errors in these cutoffs.