| 释义 |
degrees of freedom
Physics
1. The number of independent parameters required to specify the configuration of a system. This concept is applied in the kinetic theory to specify the number of independent ways in which an atom or molecule can take up energy. There are however various sets of parameters that may be chosen, and the details of the consequent theory vary with the choice. For example, in a monatomic gas each atom may be allotted three degrees of degrees of freedom, corresponding to the three coordinates in space required to specify its position. The mean energy per atom for each degree of freedom is the same, according to the principle of the equipartition of energy, and is equal to kT/2 for each degree of freedom (where k is the Boltzmann constant and T is the thermodynamic temperature). Thus for a monatomic gas the total molar energy is 3LkT/2, where L is the Avogadro constant (the number of atoms per mole). As k=R/L, where R is the molar gas constant, the total molar energy is 3RT/2. In a diatomic gas the two atoms require six coordinates between them, giving six degrees of freedom. Commonly these are interpreted as six independent ways of storing energy: on this basis the molecule has three degrees of freedom for different directions of translational motion (see translation), and in addition there are two degrees of freedom for rotation of the molecular axis and one vibrational degree of freedom along the bond between the atoms. The rotational degrees of freedom each contribute kT/2, to the total energy; similarly the vibrational degree of freedom has an equal share of kinetic energy and must on average have as much potential energy (see simple harmonic motion). The total energy per molecule for a diatomic gas is therefore 3kT/2 (for translational energy of the whole molecule) plus 2kT/2 (for rotational energy of each atom) plus 2kT/2 (for vibrational energy), i.e. a total of 7kT/2. 2. The least number of independent variables required to define the state of a system in the phase rule. In this sense a gas has two degrees of freedom (e.g. temperature and pressure).
Astronomy
The number of independent variables in a moving system such as a group of orbiting bodies. In the n-body problem, the number of degrees of freedom rises with the number of gravitating bodies involved, greatly increasing the complexity of the problem.
Statistics
A parameter that appears in some probability distributions used in statistical inference, particularly the t-distribution, the chi-squared distribution, and the F-distribution. The phrase ‘degrees of freedom’ was introduced by Sir Ronald Fisher in 1922. In the case of the t-distribution, the term usually reflects the fact that the population variance has been estimated. The number of degrees of freedom is equal to the number of independent pieces of information concerning the variance. In the most familiar case of n observations, x1, x2,…, xn, from a population with unknown mean and variance, there are (n−1) independent deviations from the mean, since , where . In this case, therefore, there are (n−1) degrees of freedom. In the case of a random variable with a chi-squared distribution, if it can be expressed as the sum of squares of m independent standard normal variables (see normal distribution), then the distribution has m degrees of freedom. In the case where the chi-squared distribution is used as a goodness-of-fit test, each independent parameter estimated from the data represents another constraint, so the number of degrees of freedom is (c−1−p), where c is the number of cells, p is the number of parameters estimated from the sample data, and there is the constraint that the sum of the observed frequencies is the sum of the expected frequencies. In the case where the chi-squared distribution is used to test the null hypothesis of independence in a J×K contingency table, there are (J−1) (K−1) degrees of freedom.
Chemistry
1. The number of independent parameters required to specify the configuration of a system. This concept is applied in the kinetic theory to specify the number of independent ways in which an atom or molecule can take up energy. There are however various sets of parameters that may be chosen, and the details of the consequent theory vary with the choice. For example, in a monatomic gas each atom may be allotted three degrees of freedom, corresponding to the three coordinates in space required to specify its position. The mean energy per atom for each degree of freedom is the same, according to the principle of the equipartition of energy, and is equal to kT/2 for each degree of freedom (where k is the Boltzmann constant and T is the thermodynamic temperature). Thus for a monatomic gas the total molar energy is 3LkT/2, where L is the Avogadro constant (the number of atoms per mole). As k = R/L, where R is the molar gas constant, the total molar energy is 3RT/2. In a diatomic gas the two atoms require six coordinates between them, giving six degrees of freedom. Commonly these are interpreted as six independent ways of storing energy: on this basis the molecule has three degrees of freedom for different directions of translational motion, and in addition there are two degrees of freedom for rotation of the molecular axis and one vibrational degree of freedom along the bond between the atoms. The rotational degrees of freedom each contribute their share, kT/2, to the total energy; similarly the vibrational degree of freedom has an equal share of kinetic energy and must on average have as much potential energy. The total energy per molecule for a diatomic gas is therefore 3kT/2 (for translational energy of the whole molecule) plus 2kT/2 (for rotational energy) plus 2kT/2 (for vibrational energy), i.e. a total of 7kT/2. 2. The least number of independent variables required to define the state of a system in the phase rule. In this sense a gas has two degrees of freedom (e.g. temperature and pressure).
Chemical Engineering
The number of independent variables that are needed to describe fully the equilibrium state of a system or process. In defining the thermodynamic equilibrium for a system involving components and phases, the thermodynamic degrees of freedom are the minimum number of variables (temperature, pressure, and composition) that must be stated to define the system completely. In the use of statistics, the degrees of freedom refers to the number of values that can be varied in a calculation determined from the difference between the number of independent values (sample size) and the steps (which is usually one). It is used frequently in linear regression and analysis of variance calculations, and in some other calculations involving the sum of squares.
Computer
In statistical analysis, the number of independent observations associated with an estimate of variance (see measures of variation) or of a component of an analysis of variance. The simplest example is in estimating the variance of a sample of n observations, where the degrees of freedom is n − 1, the divisor of the sum of squared deviations from the sample mean. More generally, the degrees of freedom, f, is the difference between the number of parameters in a given model, and a special case of that model with fewer parameters. The number of degrees of freedom is required when selecting a particular instance of the chi-squared distribution, Student’s t distribution, or F distribution.
Geology and Earth Sciences
1. When a substance is heated its kinetic energy increases. Kinetic energy is made up from the translation and rotation of particles, and the vibration of atoms which constitute the molecules of a substance. A substance may, therefore, absorb heat energy supplied to it in several ways, and is said to possess a number of degrees of freedom. In general, a molecule consisting of N atoms will have 3N degrees of freedom; thus for a diatomic molecule there will be six degrees of freedom: three will be translational, two rotational, and one vibrational. In a phase diagram, describing, for example, a three-phase system (such as ice–water–vapour), pressure and/or temperature can be altered independently, in an area where only one phase exists, without altering the one-phase condition. Along the line separating two areas, if temperature is altered then pressure must alter accordingly, or vice versa, to maintain the two-phase equilibrium. At a point where three phases are in equilibrium, alteration of either temperature or pressure will cause one phase to disappear. The system thus possesses (a) two degrees of freedom in the area; (b) one degree of freedom along the line; and (c) no degrees of freedom at the point. 2. In statistics, the number of independent variables involved in calculating a statistic. This value is equal to the difference between the total number of data points under consideration, and the number of restrictions. The number of restrictions is equal to the number of parameters which are the same in both observed data set and theoretical data set, e.g. total cumulative values, means.
Economics
The minimal number of independent characteristics, or variables, required to specify completely the state of the system at a given moment. If there exists a constraint or a set of constraints on these variables, in other words, relationships among the variables, each such constraint reduces the number of degrees of freedom. Thus, the number of degrees of freedom equals the number of variables that completely specify the system less the number of constraints imposed on these variables. For example, the number of degrees of freedom for the least squares residuals from a linear regression model with K parameters estimated from a sample of N observations equals (N −K), since in minimizing the sum of squared residuals K first-order conditions are imposed on N data points.
|