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where is a Christoffel symbol of the second kind, which contains the metric (with respect to the coordinate system). |
Given the mass-energy distribution provided by the stress–energy tensor , the Einstein field equations are a set of non-linear second-order partial differential equations in the metric, and imply the curvature of spacetime is equivalent to a gravitational field (see equivalence principle). Mass falling in curved spacetime is equivalent to a mass falling in a gravitational field - because gravity is a fictitious force. The "relative acceleration" of one geodesic to another in curved spacetime is given by the "geodesic deviation equation": |
where is the separation vector between two geodesics, ("not" just ) is the covariant derivative, and is the Riemann curvature tensor, containing the Christoffel symbols. In other words, the geodesic deviation equation is the equation of motion for masses in curved spacetime, analogous to the Lorentz force equation for charges in an electromagnetic field. |
For flat spacetime, the metric is a constant tensor so the Christoffel symbols vanish, and the geodesic equation has the solutions of straight lines. This is also the limiting case when masses move according to Newton's law of gravity. |
In general relativity, rotational motion is described by the relativistic angular momentum tensor, including the spin tensor, which enter the equations of motion under covariant derivatives with respect to proper time. The Mathisson–Papapetrou–Dixon equations describe the motion of spinning objects moving in a gravitational field. |
Unlike the equations of motion for describing particle mechanics, which are systems of coupled ordinary differential equations, the analogous equations governing the dynamics of waves and fields are always partial differential equations, since the waves or fields are functions of space and time. For a particular solution, boundary conditions along with initial conditions need to be specified. |
Sometimes in the following contexts, the wave or field equations are also called "equations of motion". |
Equations that describe the spatial dependence and time evolution of fields are called "field equations". These include |
This terminology is not universal: for example although the Navier–Stokes equations govern the velocity field of a fluid, they are not usually called "field equations", since in this context they represent the momentum of the fluid and are called the "momentum equations" instead. |
Equations of wave motion are called "wave equations". The solutions to a wave equation give the time-evolution and spatial dependence of the amplitude. Boundary conditions determine if the solutions describe traveling waves or standing waves. |
From classical equations of motion and field equations; mechanical, gravitational wave, and electromagnetic wave equations can be derived. The general linear wave equation in 3D is: |
where is any mechanical or electromagnetic field amplitude, say: |
and is the phase velocity. Nonlinear equations model the dependence of phase velocity on amplitude, replacing by . There are other linear and nonlinear wave equations for very specific applications, see for example the Korteweg–de Vries equation. |
In quantum theory, the wave and field concepts both appear. |
In quantum mechanics, in which particles also have wave-like properties according to wave–particle duality, the analogue of the classical equations of motion (Newton's law, Euler–Lagrange equation, Hamilton–Jacobi equation, etc.) is the Schrödinger equation in its most general form: |
where is the wavefunction of the system, is the quantum Hamiltonian operator, rather than a function as in classical mechanics, and is the Planck constant divided by 2. Setting up the Hamiltonian and inserting it into the equation results in a wave equation, the solution is the wavefunction as a function of space and time. The Schrödinger equation itself reduces to the Hamilton–Jacobi equation when one considers the correspondence principle, in the limit that becomes zero. |
Throughout all aspects of quantum theory, relativistic or non-relativistic, there are various formulations alternative to the Schrödinger equation that govern the time evolution and behavior of a quantum system, for instance: |
In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. The partition function is dimensionless, it is a pure number. |
Each partition function is constructed to represent a particular statistical ensemble (which, in turn, corresponds to a particular free energy). The most common statistical ensembles have named partition functions. The canonical partition function applies to a canonical ensemble, in which the system is allowed to exchange heat with the environment at fixed temperature, volume, and number of particles. The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential. Other types of partition functions can be defined for different circumstances; see partition function (mathematics) for generalizations. The partition function has many physical meanings, as discussed in Meaning and significance. |
Initially, let us assume that a thermodynamically large system is in thermal contact with the environment, with a temperature "T", and both the volume of the system and the number of constituent particles are fixed. A collection of this kind of system comprises an ensemble called a canonical ensemble. The appropriate mathematical expression for the canonical partition function depends on the degrees of freedom of the system, whether the context is classical mechanics or quantum mechanics, and whether the spectrum of states is discrete or continuous. |
For a canonical ensemble that is classical and discrete, the canonical partition function is defined as |
The exponential factor formula_7 is otherwise known as the Boltzmann factor. |
In classical mechanics, the position and momentum variables of a particle can vary continuously, so the set of microstates is actually uncountable. In "classical" statistical mechanics, it is rather inaccurate to express the partition function as a sum of discrete terms. In this case we must describe the partition function using an integral rather than a sum. For a canonical ensemble that is classical and continuous, the canonical partition function is defined as |
To make it into a dimensionless quantity, we must divide it by "h", which is some quantity with units of action (usually taken to be Planck's constant). |
For a gas of formula_15 identical classical particles in three dimensions, the partition function is |
The reason for the factorial factor "N"! is discussed below. The extra constant factor introduced in the denominator was introduced because, unlike the discrete form, the continuous form shown above is not dimensionless. As stated in the previous section, to make it into a dimensionless quantity, we must divide it by "h"3"N" (where "h" is usually taken to be Planck's constant). |
For a canonical ensemble that is quantum mechanical and discrete, the canonical partition function is defined as the trace of the Boltzmann factor: |
The dimension of formula_32 is the number of energy eigenstates of the system. |
For a canonical ensemble that is quantum mechanical and continuous, the canonical partition function is defined as |
In systems with multiple quantum states "s" sharing the same energy "Es", it is said that the energy levels of the system are degenerate. In the case of degenerate energy levels, we can write the partition function in terms of the contribution from energy levels (indexed by "j") as follows: |
where "gj" is the degeneracy factor, or number of quantum states "s" that have the same energy level defined by "Ej" = "Es". |
The above treatment applies to "quantum" statistical mechanics, where a physical system inside a finite-sized box will typically have a discrete set of energy eigenstates, which we can use as the states "s" above. In quantum mechanics, the partition function can be more formally written as a trace over the state space (which is independent of the choice of basis): |
where "Ĥ" is the quantum Hamiltonian operator. The exponential of an operator can be defined using the exponential power series. |
The classical form of "Z" is recovered when the trace is expressed in terms of coherent states |
and when quantum-mechanical uncertainties in the position and momentum of a particle |
are regarded as negligible. Formally, using bra–ket notation, one inserts under the trace for each degree of freedom the identity: |
where "x", "p" is a normalised Gaussian wavepacket centered at |
A coherent state is an approximate eigenstate of both operators formula_44 and formula_45, hence also of the Hamiltonian "Ĥ", with errors of the size of the uncertainties. If Δ"x" and Δ"p" can be regarded as zero, the action of "Ĥ" reduces to multiplication by the classical Hamiltonian, and "Z" reduces to the classical configuration integral. |
For simplicity, we will use the discrete form of the partition function in this section. Our results will apply equally well to the continuous form. |
Consider a system "S" embedded into a heat bath "B". Let the total energy of both systems be "E". Let "pi" denote the probability that the system "S" is in a particular microstate, "i", with energy "Ei". According to the fundamental postulate of statistical mechanics (which states that all attainable microstates of a system are equally probable), the probability "pi" will be proportional to the number of microstates of the total closed system ("S", "B") in which "S" is in microstate "i" with energy "Ei". Equivalently, "pi" will be proportional to the number of microstates of the heat bath "B" with energy "E" − "Ei": |
Assuming that the heat bath's internal energy is much larger than the energy of "S" ("E" ≫ "Ei"), we can Taylor-expand formula_47 to first order in "Ei" and use the thermodynamic relation formula_48, where here formula_49, formula_50 are the entropy and temperature of the bath respectively: |
Since the total probability to find the system in "some" microstate (the sum of all "pi") must be equal to 1, we know that the constant of proportionality must be the normalization constant, and so, we can define the partition function to be this constant: |
In order to demonstrate the usefulness of the partition function, let us calculate the thermodynamic value of the total energy. This is simply the expected value, or ensemble average for the energy, which is the sum of the microstate energies weighted by their probabilities: |
Incidentally, one should note that if the microstate energies depend on a parameter λ in the manner |
then the expected value of "A" is |
This provides us with a method for calculating the expected values of many microscopic quantities. We add the quantity artificially to the microstate energies (or, in the language of quantum mechanics, to the Hamiltonian), calculate the new partition function and expected value, and then set "λ" to zero in the final expression. This is analogous to the source field method used in the path integral formulation of quantum field theory. |
In this section, we will state the relationships between the partition function and the various thermodynamic parameters of the system. These results can be derived using the method of the previous section and the various thermodynamic relations. |
As we have already seen, the thermodynamic energy is |
The variance in the energy (or "energy fluctuation") is |
In general, consider the extensive variable X and intensive variable Y where X and Y form a pair of conjugate variables. In ensembles where Y is fixed (and X is allowed to fluctuate), then the average value of X will be: |
The sign will depend on the specific definitions of the variables X and Y. An example would be X = volume and Y = pressure. Additionally, the variance in X will be |
In the special case of entropy, entropy is given by |
where "A" is the Helmholtz free energy defined as "A" = "U" − "TS", where "U" = "E" is the total energy and "S" is the entropy, so that |
Furthermore, the heat capacity can be expressed as |
Suppose a system is subdivided into "N" sub-systems with negligible interaction energy, that is, we can assume the particles are essentially non-interacting. If the partition functions of the sub-systems are "ζ"1, "ζ"2, ..., "ζ"N, then the partition function of the entire system is the "product" of the individual partition functions: |
If the sub-systems have the same physical properties, then their partition functions are equal, ζ1 = ζ2 = ... = ζ, in which case |
However, there is a well-known exception to this rule. If the sub-systems are actually identical particles, in the quantum mechanical sense that they are impossible to distinguish even in principle, the total partition function must be divided by a "N"! ("N" factorial): |
This is to ensure that we do not "over-count" the number of microstates. While this may seem like a strange requirement, it is actually necessary to preserve the existence of a thermodynamic limit for such systems. This is known as the Gibbs paradox. |
It may not be obvious why the partition function, as we have defined it above, is an important quantity. First, consider what goes into it. The partition function is a function of the temperature "T" and the microstate energies "E"1, "E"2, "E"3, etc. The microstate energies are determined by other thermodynamic variables, such as the number of particles and the volume, as well as microscopic quantities like the mass of the constituent particles. This dependence on microscopic variables is the central point of statistical mechanics. With a model of the microscopic constituents of a system, one can calculate the microstate energies, and thus the partition function, which will then allow us to calculate all the other thermodynamic properties of the system. |
The partition function can be related to thermodynamic properties because it has a very important statistical meaning. The probability "Ps" that the system occupies microstate "s" is |
Thus, as shown above, the partition function plays the role of a normalizing constant (note that it does "not" depend on "s"), ensuring that the probabilities sum up to one: |
This is the reason for calling "Z" the "partition function": it encodes how the probabilities are partitioned among the different microstates, based on their individual energies. The letter "Z" stands for the German word "Zustandssumme", "sum over states". The usefulness of the partition function stems from the fact that it can be used to relate macroscopic thermodynamic quantities to the microscopic details of a system through the derivatives of its partition function. Finding the partition function is also equivalent to performing a Laplace transform of the density of states function from the energy domain to the β domain, and the inverse Laplace transform of the partition function reclaims the state density function of energies. |
We can define a grand canonical partition function for a grand canonical ensemble, which describes the statistics of a constant-volume system that can exchange both heat and particles with a reservoir. The reservoir has a constant temperature "T", and a chemical potential "μ". |
The grand canonical partition function, denoted by formula_71, is the following sum over microstates |
Here, each microstate is labelled by formula_73, and has total particle number formula_74 and total energy formula_75. This partition function is closely related to the grand potential, formula_76, by the relation |
This can be contrasted to the canonical partition function above, which is related instead to the Helmholtz free energy. |
It is important to note that the number of microstates in the grand canonical ensemble may be much larger than in the canonical ensemble, since here we consider not only variations in energy but also in particle number. Again, the utility of the grand canonical partition function is that it is related to the probability that the system is in state formula_73: |
An important application of the grand canonical ensemble is in deriving exactly the statistics of a non-interacting many-body quantum gas (Fermi–Dirac statistics for fermions, Bose–Einstein statistics for bosons), however it is much more generally applicable than that. The grand canonical ensemble may also be used to describe classical systems, or even interacting quantum gases. |
The grand partition function is sometimes written (equivalently) in terms of alternate variables as |
where formula_81 is known as the absolute activity (or fugacity) and formula_82 is the canonical partition function. |
In combustion, the Williams spray equation, also known as the Williams–Boltzmann equation, describes the statistical evolution of sprays contained in another fluid, analogous to the Boltzmann equation for the molecules, named after Forman A. Williams, who derived the equation in 1958. |
The sprays are assumed to be spherical with radius formula_1, even though the assumption is valid for solid particles(liquid droplets) when their shape has no consequence on the combustion. For liquid droplets to be nearly spherical, the spray has to be dilute(total volume occupied by the sprays is much less than the volume of the gas) and the Weber number formula_2, where formula_3 is the gas density, formula_4 is the spray droplet velocity, formula_5 is the gas velocity and formula_6 is the surface tension of the liquid spray, should be formula_7. |
The equation is described by a number density function formula_8, which represents the probable number of spray particles (droplets) of chemical species formula_9 (of formula_10 total species), that one can find with radii between formula_1 and formula_12, located in the spatial range between formula_13 and formula_14, traveling with a velocity in between formula_4 and formula_16, having the temperature in between formula_17 and formula_18 at time formula_19. Then the spray equation for the evolution of this density function is given by |
A simplified model for liquid propellant rocket. |
This model for the rocket motor was developed by Probert, Williams and Tanasawa. It is reasonable to neglect formula_31, for distances not very close to the spray atomizer, where major portion of combustion occurs. Consider a one-dimensional liquid-propellent rocket motor situated at formula_32, where fuel is sprayed. Neglecting formula_33(density function is defined without the temperature so accordingly dimensions of formula_34 changes) and due to the fact that the mean flow is parallel to formula_35 axis, the steady spray equation reduces to |
where formula_37 is the velocity in formula_35 direction. Integrating with respect to the velocity results |
The contribution from the last term (spray acceleration term) becomes zero (using Divergence theorem) since formula_40 when formula_41 is very large, which is typically the case in rocket motors. The drop size rate formula_42 is well modeled using vaporization mechanisms as |
where formula_44 is independent of formula_1, but can depend on the surrounding gas. Defining the number of droplets per unit volume per unit radius and average quantities averaged over velocities, |
If further assumed that formula_48 is independent of formula_1, and with a transformed coordinate |
If the combustion chamber has varying cross-section area formula_51, a known function for formula_52 and with area formula_53 at the spraying location, then the solution is given by |
where formula_55 are the number distribution and mean velocity at formula_32 respectively. |
The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stated by Benoît Paul Émile Clapeyron in 1834 as a combination of the empirical Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. The ideal gas law is often written in an empirical form: |
where formula_2, formula_3 and formula_4 are the pressure, volume and temperature; formula_5 is the amount of substance; and formula_6 is the ideal gas constant. It is the same for all gases. |
It can also be derived from the microscopic kinetic theory, as was achieved (apparently independently) by August Krönig in 1856 and Rudolf Clausius in 1857. |
Note that this law makes no comment as to whether a gas heats or cools during compression or expansion. An ideal gas may not change temperature, but most gases like air are not ideal and follow the Joule–Thomson effect. |
The state of an amount of gas is determined by its pressure, volume, and temperature. The modern form of the equation relates these simply in two main forms. The temperature used in the equation of state is an absolute temperature: the appropriate SI unit is the kelvin. |
In SI units, "p" is measured in pascals, "V" is measured in cubic metres, "n" is measured in moles, and "T" in kelvins (the Kelvin scale is a shifted Celsius scale, where 0.00 K = −273.15 °C, the lowest possible temperature). "R" has the value 8.314 J/(K·mol) ≈ 2 cal/(K·mol), or 0.0821 l·atm/(mol·K). |
How much gas is present could be specified by giving the mass instead of the chemical amount of gas. Therefore, an alternative form of the ideal gas law may be useful. The chemical amount ("n") (in moles) is equal to total mass of the gas ("m") (in kilograms) divided by the molar mass ("M") (in kilograms per mole): |
By replacing "n" with "m"/"M" and subsequently introducing density "ρ" = "m"/"V", we get: |
Defining the specific gas constant "R"specific(r) as the ratio "R"/"M", |
This form of the ideal gas law is very useful because it links pressure, density, and temperature in a unique formula independent of the quantity of the considered gas. Alternatively, the law may be written in terms of the specific volume "v", the reciprocal of density, as |
It is common, especially in engineering and meteorological applications, to represent the specific gas constant by the symbol "R". In such cases, the universal gas constant is usually given a different symbol such as formula_21 or formula_22 to distinguish it. In any case, the context and/or units of the gas constant should make it clear as to whether the universal or specific gas constant is being referred to. |
In statistical mechanics the following molecular equation is derived from first principles |
where is the absolute pressure of the gas, is the number density of the molecules (given by the ratio = , in contrast to the previous formulation in which is the "number of moles"), is the absolute temperature, and is the Boltzmann constant relating temperature and energy, given by: |
From this we notice that for a gas of mass , with an average particle mass of times the atomic mass constant, , (i.e., the mass is u) the number of molecules will be given by |
and since , we find that the ideal gas law can be rewritten as |
In SI units, is measured in pascals, in cubic metres, in kelvins, and |
Combining the laws of Charles, Boyle and Gay-Lussac gives the combined gas law, which takes the same functional form as the ideal gas law save that the number of moles is unspecified, and the ratio of formula_27 to formula_4 is simply taken as a constant: |
where formula_2 is the pressure of the gas, formula_3 is the volume of the gas, formula_4 is the absolute temperature of the gas, and formula_33 is a constant. When comparing the same substance under two different sets of conditions, the law can be written as |
According to assumptions of the kinetic theory of ideal gases, we assume that there are no intermolecular attractions between the molecules of an ideal gas. In other words, its potential energy is zero. Hence, all the energy possessed by the gas is in the kinetic energy of the molecules of the gas. |
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