# Statistical Mechanics/The Foundations

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# Introduction

The goal of statistical mechanics is to bridge the gap that exists between the microscopic and the macroscopic world. Take for example the equation governing the behaviour of microscopic particles: the quantum mechanical Schrödinger equation. Based on this equation, a quantum physicist will tell you everything you want to know about a particle by itself. If you ask, he will even be able to give you a solution for the wave function of a system of two particles. However, once you add a third or more, things start to get more complicated. There is no longer an analytic solution, and one must turn to computers to solve the problem numerically - and the results the computer will spit out are quite accurate for systems of 3, 4 or more particles. Even when considering classical Hamiltonian mechanics, there is no analytical solution for many-body problems such as the motion of the planets, although we can simulate them accurately numerically.

But what happens when the number of particles you have is much larger - not just ten or twenty, or even a thousand - what happens when you have a cup of water for example, with ~1025 particles? Each of the 1025 particles interacts with every single one of the 1025 other particles - that's a total number of interactions of the order of 1050 that would have to be computed at every instant! Even a computer that could perform a trillion calculations per second would take 1020 times longer than the age of the universe to compute the exact state of your cup of water for a single instant in time. Clearly such a computation is not possible. It is therefore not possible, in practice, to solve the equations governing macroscopic systems.

Statistical mechanics provides the tools required to take the information given by quantum physics and use it to describe macroscopic systems and predict how they will evolve in time. By far the most important of these tools is probability theory. Instead of saying that a physical system is in exactly one or the other configuration, we will talk about the probability of it being in a certain configuration. For example, in a room filled with gas, it is far more probable that the gas is spread evenly rather than being bunched up in one corner. This may seem to be nothing more than common sense, but it has profound implications, especially when the probabilities involved are studied quantitatively. Through the use of this and other tools, statistical mechanics enables physicists to gain fundamental insight into the workings of the macroscopic world.