Q: What is a trajectory?

A trajectory is the time-ordered sequence of system configurations which occur as all the coordinates evolve in time following some rules – hopefully rules embodying reasonable physical dynamics, such as Newton’s laws or constant-temperature molecular dynamics.

Q: What is a trajectory ensemble?

It’s a set of independent trajectories that together characterize a particular condition such as equilibrium or a non-equilibrium steady state.  That is, the trajectories do not interact in any way, but statistically they describe some condition because of how they have been initiated – and when they are observed relative to their initialization … see below.

Continue reading

The Markov model, without question, is one of the most powerful and elegant tools available in many fields of biological modeling and beyond.  In my world of molecular simulation, Markov models have provided analyses more insightful than would be possible with direct simulation alone.  And I’m a user, too.  Markov models, in their chemical-kinetics guise, play a prominent role in illustrating cellular biophysics in my online book, Physical Lens on the Cell.

Yet it’s fair to say that everything is Markovian and nothing is Markovian – and we need to understand this.

If you’re new to the business, a quick word on what “Markovian” means.  A Markov process is a stochastic process where the future (i.e., the distribution of future outcomes) depends only on the present state of the system.  Good examples would be chemical kinetics models with transition probabilities governed by rate constants or simple Monte Carlo simulation (a.k.a. Markov-chain Monte Carlo).  To determine the next state of the system, we don’t care about the past: only the present state matters.

Continue reading