Statistical physics

Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and in particular mathematical tools to deal with large populations and approximations, to solve physical problems. It can describe a wide variety of fields that are inherently stochastic in nature. Its applications include many problems in the fields of physics, biology, chemistry, neuroscience. Its main goal is to clarify the properties of matter as a whole, in terms of the physical laws that govern atomic motion.

Statistical mechanics develops the phenomenological results of thermodynamics from a probabilistic examination of the underlying microscopic systems. Historically, one of the first topics in physics where statistical methods were applied was the field of classical mechanics, which deals with the motion of particles or objects when subjected to a force.

Scope

Statistical physics quantitatively explains and describes superconductivity, superfluidity, turbulence, collective phenomena in solids and plasma, and structural features of liquids. It underlies modern astrophysics. In solid state physics, statistical physics aids the study of liquid crystals, phase transitions, and critical phenomena. Many experimental studies of the subject are based entirely on the statistical description of a system. These include the scattering of cold neutrons, X-rays, visible light, and more. Statistical physics also plays a role in materials science, nuclear physics, astrophysics, chemistry, biology, and medicine (for example, the study of the spread of infectious diseases).

Statistical Mechanics

Statistical mechanics provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties of materials that can be observed in everyday life, thus explaining thermodynamics as a natural result of statistics, classical mechanics and quantum mechanics at the microscopic level. Because of this history, statistical physics is often considered synonymous with statistical mechanics or statistical thermodynamics.

One of the most important equations of statistical mechanics (similar to F=ma{displaystyle F=ma} in Newtonian mechanics, or Schrödinger's equation in quantum mechanics) is the definition of partition function Z{displaystyle Z}which is essentially a weighted sum of all possible states q{displaystyle q} available for a system.

Z=␡ ␡ qe− − E(q)kBT{displaystyle Z=sum _{q}mathrm {e} ^{-{frac {E(q)}{k_{B}T}}}}}}}

where kB{displaystyle k_{B}} It's Boltzmann's constant, T{displaystyle T} It's the temperature and E(q){displaystyle E(q)} It's state energy. q{displaystyle q}. In addition, the probability of a given state, q{displaystyle q}which happens is given by

P(q)=e− − E(q)kBTZ{displaystyle P(q)={frac {mathrm {e} ^{-{frac {E(q)}{k_{B}T}}}}}{Z}}}}}}}}

Here we see that very high-energy states are unlikely to occur, a result that is consistent with intuition.

A statistical approach can work well in classical systems when the number of degrees of freedom (and thus the number of variables) is so large that an exact solution is not possible or really useful. Statistical mechanics can also describe work in nonlinear dynamics, chaos theory, thermal physics, fluid dynamics (particularly at high Knudsen numbers), or plasma physics.

Quantum Statistical Mechanics

Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics, a statistical ensemble (probability distribution over possible quantum states) is described by a density operator S, which is a self-adjoint, non-negative trace class operator of trace 1 in the Hilbert space H that describes the quantum system. This can be shown under various mathematical formalisms for quantum mechanics. One of those formalisms is provided by quantum logic.

Monte Carlo method

Although some problems in statistical physics can be solved analytically by approximation and expansion, most current research uses the great processing power of modern computers to simulate or approximate solutions. A common approach to statistical problems is to use a Monte Carlo simulation to obtain information about the properties of a complex system. Monte Carlo methods are important in computational physics, physical chemistry, and related fields, and have a variety of applications, including medical physics, where they are used to model radiation transport for radiation dosimetry calculations.

Further reading

• Reif, F. (2009). Fundamentals of Statistical and Thermal Physics. Waveland Press. ISBN 978-1-4786-1005-2.
• Müller-Kirsten, Harald J.W. (2013). Basics of Statistical Physics (2nd edition). World Scientific. ISBN 978-981-4449-55-7. doi:10.1142/8709.
• Kadanoff, Leo P. «Statistical Physics and other resources».
• Kadanoff, Leo P. (2000). Statistical Physics: Statics, Dynamics and Renormalization. World Scientific. ISBN 978-981-02-3764-6.