Question

what is configuration space and phase space? what are its coordinates. when do we use it.

Answer 1

Configurations space is a space with one axis for each positional degree of freedom in a system. So, for example, the configuration space for a single unconstrained particle has three axes, corresponding to the x, y and z degrees of freedom of the particle. The configuration space for two particles has six axes, corresponding to x1, y1, z1, x2, y2, and z2. The configuration space for N particles has 3N axes. On the other hand, the configuration space for a particle confined to move on a ring has only one axis, call it theta, since this particle can only move in one dimension.

Phase space is a space with one axis for each degree of freedom in a system. Generally that includes velocity as well as positional degrees of freedom. So, for example, the phase space of a single unconstrained particle has six axes: x, y, z, vx, vy, and vz. (Usually we actually use momentum instead of velocity, because in classical mechanics momentum is "conjugate" to position.) The phase space for N particles has 6N axes, and the phase space for a particle confined to a ring would have 2 axes.

We use phase space extensively in developing theories of classical statistical mechanics, the mathematical modeling of systems of very large numbers of degrees of freedom, where we hope to extract the time evolution of "important" collective degrees of freedom (for example energy, total momentum, total angular momentum, density, temperature, pressure, volume) even though we do not, and cannot, have a complete description of the location and velocity of each particle in the system. Roughly speaking, we define blobbish kinds of functions in phase space that describe the *probability* that the system is at one the points in phase space, and then seek to understand how that blob evolves, and its properties, say in equilibrium.

Configuration space is more often used in quantum statistical mechanics, because in quantum mechanics you cannot simultaneously know position and momentum, so the usefulness of phase space considerably declines. Quantum stat mech has the same goals of classical stat mech, but it is used in situations where the underlying quantum mechanics of the system are important. The most important such case is probably electrons in delocalized systems, such as metals. The electrons in such systems are strongly quantum mechanical, and quantum stat mech is necessary to achieve any understandign of their behaviour. Other applications include superconductivity or superfluidity, or the behaviour of neutrons stars.