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Hypernetted Chain Equation
In statistical mechanics the hypernetted-chain equation is a closure relation to solve the Ornstein–Zernike equation which relates the direct correlation function to the total correlation function. It is commonly used in fluid theory to obtain e.g. expressions for the radial distribution function. It is given by: : \ln y(r_) =\ln g(r_) + \beta u(r_) =\rho \int \left (r_) - \ln g(r_) - \beta u(r_)\righth(r_) \, d \mathbf, \, where \rho = \frac is the number density of molecules, h(r) = g(r)-1, g(r) is the radial distribution function, u(r) is the direct interaction between pairs. \beta = \frac with T being the Thermodynamic temperature and k_ the Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the .... Derivation The direct correlation function represen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in a wide variety of fields such as biology, neuroscience, computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ..., information theory and sociology. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics arose out of the development of classical thermodynamics, a field for which it was successful in explaining macroscopic physical properties—such as temperature, pressure, and heat capacity—in terms of microscop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure (mathematics)
In mathematics, a subset of a given set (mathematics), set is closed under an Operation (mathematics), operation on the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: is not a natural number, although both 1 and 2 are. Similarly, a subset is said to be closed under a ''collection'' of operations if it is closed under each of the operations individually. The closure of a subset is the result of a closure operator applied to the subset. The ''closure'' of a subset under some operations is the smallest superset that is closed under these operations. It is often called the ''span'' (for example linear span) or the ''generated set''. Definitions Let be a set (mathematics), set equipped with one or several methods for producing elements of from other elements of .Operation (mathematics), Operations and (partial function, partial) multivar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ornstein–Zernike Equation
In statistical mechanics the Ornstein–Zernike (OZ) equation is an integral equation introduced by Leonard Ornstein and Frits Zernike that relates different correlation functions with each other. Together with a closure relation, it is used to compute the structure factor and thermodynamic state functions of amorphous matter like liquids or colloids. Context The OZ equation has practical importance as a foundation for approximations for computing the pair correlation function of molecules or ions in liquids, or of colloidal particles. The pair correlation function is related via Fourier transform to the static structure factor, which can be determined experimentally using X-ray diffraction or neutron diffraction. The OZ equation relates the pair correlation function to the direct correlation function. The direct correlation function is only used in connection with the OZ equation, which can actually be seen as its definition. Besides the OZ equation, other methods for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radial Distribution Function
In statistical mechanics, the radial distribution function, (or pair correlation function) g(r) in a system of particles (atoms, molecules, colloids, etc.), describes how density varies as a function of distance from a reference particle. If a given particle is taken to be at the origin ''O'', and if \rho = N/V is the average number density of particles, then the local time-averaged density at a distance r from ''O'' is \rho g(r). This simplified definition holds for a homogeneous and isotropic system. A more general case will be considered below. In simplest terms it is a measure of the probability of finding a particle at a distance of r away from a given reference particle, relative to that for an ideal gas. The general algorithm involves determining how many particles are within a distance of r and r+dr away from a particle. This general theme is depicted to the right, where the red particle is our reference particle, and the blue particles are those whose centers are with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thermodynamic Temperature
Thermodynamic temperature, also known as absolute temperature, is a physical quantity which measures temperature starting from absolute zero, the point at which particles have minimal thermal motion. Thermodynamic temperature is typically expressed using the Kelvin scale, where the unit of measurement is the ''kelvin'' (unit symbol: K). The Kelvin scale uses the same degree interval as the Celsius scale but is offset so that 0 K corresponds to absolute zero. For comparison, a temperature of 295 K corresponds to 21.85 °C and 71.33 °F. Another absolute scale of temperature is the Rankine scale, which is based on the Fahrenheit degree interval. Historically, thermodynamic temperature was defined by Lord Kelvin in terms of a macroscopic relation between Work (thermodynamics), thermodynamic work and Heat, heat transfer as defined in thermodynamics, but the kelvin was redefined by international agreement in 2019 in terms of phenomena that are now understood as man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boltzmann Constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the molar gas constant, in Planck's law of black-body radiation and Boltzmann's entropy formula, and is used in calculating Johnson–Nyquist noise, thermal noise in resistors. The Boltzmann constant has Dimensional analysis, dimensions of energy divided by temperature, the same as entropy and heat capacity. It is named after the Austrian scientist Ludwig Boltzmann. As part of the 2019 revision of the SI, the Boltzmann constant is one of the seven "Physical constant, defining constants" that have been defined so as to have exact finite decimal values in SI units. They are used in various combinations to define the seven SI base units. The Boltzmann constant is defined to be exactly joules per kelvin, with the effect of defining the SI unit ke ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Potential Of Mean Force
When examining a system computationally one may be interested in knowing how the free energy changes as a function of some inter- or intramolecular coordinate (such as the distance between two atoms or a torsional angle). The free energy surface along the chosen coordinate is referred to as the potential of mean force (PMF). If the system of interest is in a solvent, then the PMF also incorporates the solvent effects. General description The PMF can be obtained in Monte Carlo or molecular dynamics Molecular dynamics (MD) is a computer simulation method for analyzing the Motion (physics), physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamics ( ... simulations to examine how a system's energy changes as a function of some specific reaction coordinate parameter. For example, it may examine how the system's energy changes as a function of the distance between two residues, or as a protei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical-map Hypernetted-chain Method
The classical-map hypernetted-chain method (CHNC method) is a method used in many-body theoretical physics for interacting uniform electron liquids in two and three dimensions, and for non-ideal plasmas. The method extends the famous hypernetted-chain method (HNC) introduced by J.M.J. van Leeuwen et al. to quantum fluids as well. The classical HNC, together with the Percus–Yevick approximation, are the two pillars which bear the brunt of most calculations in the theory of interacting classical fluids. Also, HNC and PY have become important in providing basic reference schemes in the theory of fluids, and hence they are of great importance to the physics of many-particle systems. The HNC and PY integral equations provide the pair distribution functions of the particles in a classical fluid, even for very high coupling strengths. The coupling strength is measured by the ratio of the potential energy to the kinetic energy. In a classical fluid, the kinetic energy is proportion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Percus–Yevick Approximation
In statistical mechanics the Percus–Yevick approximation is a Closure (mathematics), closure relation to solve the Ornstein–Zernike equation. It is also referred to as the Percus–Yevick equation. It is commonly used in fluid theory to obtain e.g. expressions for the radial distribution function. The approximation is named after Jerome K. Percus and George J. Yevick. Derivation The direct correlation function represents the direct correlation between two particles in a system containing ''N'' − 2 other particles. It can be represented by : c(r)=g_(r) - g_(r) \, where g_(r) is the radial distribution function, i.e. g(r)=\exp[-\beta w(r)] (with ''w''(''r'') the potential of mean force) and g_(r) is the radial distribution function without the direct interaction between pairs u(r) included; i.e. we write g_(r)=\exp[-\beta(w(r)-u(r))]. Thus we ''approximate'' ''c''(''r'') by : c(r)=e^- e^. \, If we introduce the function y(r)=e^g(r) into the approximation for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
