2009 Student Research Conference:

22nd Annual Student Research Conference

22nd Annual Student Research Conference

**Random Walks in Phase Space**

Adam J. Vogt

Prof. Taner Edis, Faculty Mentor

Markov random walks can be used to describe the memory-less trajectory of a classical particle's state in phase space. Each random step in the trajectory is subject to a particular transition probability distribution function (PDF), and in order to describe the trajectory over a continuous space, the limit is taken for infinitesimal steps over an infinitesimal time interval. The limit results in a partial differential equation describing the time evolution of the probability distribution of states (PDS) for the particle. The PDS gives the probability distribution of being in a particular state at a particular time during the random walk. The existence of long-time attractors for the PDS in which the particle's state follows a preferred path in phase space would correspond to particular energy states to which the particle is constrained. This project studies random walks using various transition PDFs and describes the long-time attractors for each corresponding PDS.

**Keywords:** random walk, phase space, classical mechanics

**Topic(s):**Physics

**Presentation Type:** Oral Paper

**Session:** 22-3

**Location:** VH 1412

**Time:** 10:00