A map of global long term monthly average surface air temperatures in Mollweide projection.

Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot. Heat spontaneously flows from bodies of a higher temperature to bodies of lower temperature, at a rate that increases with the temperature difference and the thermal conductivity. No heat will be exchanged between bodies of the same temperature; such bodies are said to be in "thermal equilibrium".

The temperature of a substance typically varies with the average speed of the particles that it contains, raised to the second power; that is, it is proportional to the mean kinetic energy of its constituent particles. Formally, temperature is defined as the derivative of the internal energy with respect to the entropy.

Quantitatively, temperature is measured with thermometers, which may be calibrated to a variety of temperature scales.

Thermal vibration of a segment of protein alpha helix. The amplitude of the vibrations increases with temperature.

Temperature plays an important role in all fields of natural science, including physics, geology, chemistry, atmospheric sciences and biology.

Use in science[link]

Annual mean temperature around the world

Many physical properties of materials including the phase solid, liquid, gaseous or plasma, density, solubility, vapor pressure, and electrical conductivity depend on the temperature. Temperature also plays an important role in determining the rate and extent to which chemical reactions occur. This is one reason why the human body has several elaborate mechanisms for maintaining the temperature at 310 K, since temperatures only a few degrees higher can result in harmful reactions with serious consequences. Temperature also determines the thermal radiation emitted from a surface. One application of this effect is the incandescent light bulb, in which a tungsten filament is electrically heated to a temperature at which significant quantities of visible light are emitted.

Temperature scales[link]

Much of the world uses the Celsius scale (°C) for most temperature measurements.^{[citation needed]} It has the same incremental scaling as the Kelvin scale used by scientists, but fixes its null point, at 0°C = 273.15K, approximately the freezing point of water (under one atmosphere of pressure).^{[note 1]} The United States uses the Fahrenheit scale for common purposes, a scale on which water freezes at 32 °F and boils at 212 °F (under one atmosphere of pressure).^{[citation needed]}

For practical purposes of scientific temperature measurement, the International System of Units (SI) defines a scale and unit for the thermodynamic temperature by using the easily reproducible temperature of the triple point of water as a second reference point. The reason for this choice is that, unlike the freezing and boiling point temperatures, the temperature at the triple point is independent of pressure (since the triple point is a fixed point on a two-dimensional plot of pressure vs. temperature). For historical reasons, the triple point temperature of water is fixed at 273.16 units of the measurement increment, which has been named the kelvin in honor of the Scottish physicist who first defined the scale. The unit symbol of the kelvin is K.

Absolute zero is defined as a temperature of precisely 0 kelvins, which is equal to −273.15 °C or −459.68 °F.

Thermodynamic approach to temperature[link]

Temperature is one of the principal quantities studied in the field of thermodynamics. Thermodynamics investigates the relation between heat and work, using a special scale of temperature called the absolute temperature, and thus relates temperature to work, as considered below. In thermodynamic terms, temperature is a macroscopic intensive variable because it is independent of the bulk amount of elementary entities contained inside, be they atoms, molecules, or electrons. Real world systems are not homogeneous. For study, a body is usually spatially and temporally divided conceptually into imagined 'cells' of small size. If classical thermodynamic equilibrium conditions for matter are fulfilled to good approximation in each 'cell', then a temperature exists for each 'cell', and local thermodynamic equilibrium is said to prevail in the body.

Statistical mechanics approach to temperature[link]

Statistical mechanics provides a microscopic explanation of temperature, based on macroscopic systems' being composed of many particles, such as molecules and ions of various species, the particles of a species being all alike. It explains macroscopic phenomena in terms of the mechanics of the molecules and ions, and statistical assessments of their joint adventures. In the statistical thermodynamic approach, degrees of freedom are used instead of particles.

On the molecular level, temperature is the result of the motion of the particles that constitute the material. Moving particles carry kinetic energy. Temperature increases as this motion and the kinetic energy increase. The motion may be the translational motion of particles, or the energy of the particle due to molecular vibration or the excitation of an electron energy level. Although very specialized laboratory equipment is required to directly detect the translational thermal motions, thermal collisions by atoms or molecules with small particles suspended in a fluid produces Brownian motion that can be seen with an ordinary microscope. The thermal motions of atoms are very fast and temperatures close to absolute zero are required to directly observe them. For instance, when scientists at the NIST achieved a record-setting low temperature of 700 nK (1 nK = 10⁻⁹ K) in 1994, they used laser equipment to create an optical lattice to adiabatically cool caesium atoms. They then turned off the entrapment lasers and directly measured atom velocities of 7mm per second in order to calculate their temperature.

Molecules, such as oxygen (O₂), have more degrees of freedom than single spherical atoms: they undergo rotational and vibrational motions as well as translations. Heating results in an increase in temperature due to an increase in the average translational energy of the molecules. Heating will also cause, through equipartitioning, the energy associated with vibrational and rotational modes to increase. Thus a diatomic gas will require a higher energy input to increase its temperature by a certain amount, i.e. it will have a higher heat capacity than a monatomic gas.

The process of cooling involves removing thermal energy from a system. When no more energy can be removed, the system is at absolute zero, which cannot be achieved experimentally. Absolute zero is the null point of the thermodynamic temperature scale, also called absolute temperature. If it were possible to cool a system to absolute zero, all motion of the particles comprising matter would cease and they would be at complete rest in this classical sense. Microscopically in the description of quantum mechanics, however, matter still has zero-point energy even at absolute zero, because of the uncertainty principle.

Basic theory[link]

As distinct from a quantity of heat, temperature may be viewed as a measure of a quality of a body ^[1] or of heat.^[2][3][4][5] The quality is called hotness by some writers.^[6][7]

When two systems are at the same temperature, no net heat transfer occurs spontanteously, by conduction or radiation, between them. When a temperature difference does exist, and there is a thermally conductive or radiative connection between them, there is spontaneous heat transfer from the warmer system to the colder system, until they are at mutual thermal equilibrium. Heat transfer occurs by conduction or by thermal radiation.^{[8][9][10][11][12][13][14][15]}

Experimental physicists, for example Galileo and Newton,^[16] found that there are indefinitely many empirical temperature scales.

Temperature for bodies in thermodynamic equilibrium[link]

For experimental physics, hotness means that, when comparing any two given bodies in their respective separate thermodynamic equilibria, any two suitably given empirical thermometers with numerical scale readings will agree as to which is the hotter of the two given bodies, or that they have the same temperature.^[17] This does not require the two thermometers to have a linear relation between their numerical scale readings, but it does require that the relation between their numerical readings shall be strictly monotonic.^[18][19] A definite sense of greater hotness can be had, independently of calorimetry, of thermodynamics, and of properties of particular materials, from Wien's displacement law of thermal radiation: the temperature of a bath of thermal radiation is proportional, by a universal constant, to the frequency of the maximum of its frequency spectrum; this frequency is always positive, but can have values that tend to zero. Thermal radiation is initially defined for a cavity in thermodynamic equilibrium. These physical facts justify a mathematical statement that hotness exists on an ordered one-dimensional manifold. This is a fundamental character of temperature and thermometers for bodies in their own thermodynamic equilibrium.^{[20][6][21][22][7]}

Except for a system undergoing a first-order phase change such as the melting of ice, as a closed system receives heat, without change in its volume and without change in external force fields acting on it, its temperature rises. For a system undergoing such a phase change so slowly that departure from thermodynamic equilibrium can be neglected, its temperature remains constant as the system is supplied with latent heat. Conversely, a loss of heat from a closed system, without phase change, without change of volume, and without change in external force fields acting on it, decreases its temperature.^[23]

Temperature for bodies in a steady state but not in thermodynamic equilibrium[link]

While for bodies in their own thermodynamic equilibrium states, the notion of temperature safely requires that all empirical thermometers must agree as to which of two bodies is the hotter or that they are at the same temperature, this requirement is not safe for bodies that are in steady states though not in thermodynamic equilibrium. It can then well be that different empirical thermometers disagree about which is the hotter, and if this is so, then at least one of the bodies does not have a well defined absolute thermodynamic temperature. Nevertheless, any one given body and any one suitable empirical thermometer can still support notions of empirical, non-absolute, hotness and temperature, for a suitable range of processes. This is a matter for study in non-equilibrium thermodynamics.

Temperature for bodies not in a steady state[link]

When a body is not in a steady state, then the notion of temperature becomes even less safe than for a body in a steady state not in thermodynamic equilibrium. This is also a matter for study in non-equilibrium thermodynamics.

Thermodynamic equilibrium axiomatics[link]

For axiomatic treatment of thermodynamic equilibrium, since the 1930's, it has become customary to refer to a zeroth law of thermodynamics. The customarily stated minimalist version of such a law postulates only that all bodies, which when thermally connected would be in thermal equilibrium, should be said to have the same temperature by definition, but by itself does not establish temperature as a quantity expressed as a real number on a scale. A more physically informative version of such a law views empirical temperature as a chart on a hotness manifold.^[6][24][22] While the zeroth law permits the definitions of many different empirical scales of temperature, the second law of thermodynamics selects the definition of a single preferred, absolute temperature, unique up to an arbitrary scale factor, whence called the thermodynamic temperature.^{[25][26][6][27][21][28]} If internal energy is considered as a function of the volume and entropy of a homogeneous system in thermodynamic equilibrium, thermodynamic absolute temperature appears as the partial derivative of internal energy with respect the entropy at constant volume. Its natural, intrinsic origin or null point is absolute zero at which the entropy of any system is at a minimum. Although this is the lowest absolute temperature described by the model, the third law of thermodynamics postulates that absolute zero cannot be attained by any physical system.

Heat capacity[link]

When a sample is heated, meaning it receives thermal energy from an external source, some of the introduced heat is converted into kinetic energy, the rest to other forms of internal energy, specific to the material. The amount converted into kinetic energy causes the temperature of the material to rise. The introduced heat (Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta Q ) divided by the observed temperature change is the heat capacity (C) of the material.

Failed to parse (Missing texvc executable; please see math/README to configure.): C = \frac{\Delta Q}{\Delta T}

If heat capacity is measured for a well defined amount of substance, the specific heat is the measure of the heat required to increase the temperature of such a unit quantity by one unit of temperature. For example, to raise the temperature of water by one kelvin (equal to one degree Celsius) requires 4186 joules per kilogram (J/kg)..

Temperature measurement[link]

A typical Celsius thermometer measures a winter day temperature of -17°C.

Temperature measurement using modern scientific thermometers and temperature scales goes back at least as far as the early 18th century, when Gabriel Fahrenheit adapted a thermometer (switching to mercury) and a scale both developed by Ole Christensen Rømer. Fahrenheit's scale is still in use in the United States for non-scientific applications.

Temperature is measured with thermometers that may be calibrated to a variety of temperature scales. In most of the world (except for Belize, Myanmar, Liberia and the United States), the Celsius scale is used for most temperature measuring purposes. Most scientist measures temperature using the Celsius scale and the thermodynamic temperature using the Kelvin scale, which is the Celsius scale offset so that its null point is 0K = −273.15°C, or absolute zero. Many engineering fields in the U.S., notably high-tech and US federal specifications (civil and military), also use the Kelvin and Celsius scales. Other engineering fields in the U.S. also rely upon the Rankine scale (a shifted Fahrenheit scale) when working in thermodynamic-related disciplines such as combustion.

Units[link]

The basic unit of temperature in the International System of Units (SI) is the kelvin. It has the symbol K.

For everyday applications, it is often convenient to use the Celsius scale, in which 0°C corresponds very closely to the freezing point of water and 100°C is its boiling point at sea level. Because liquid droplets commonly exist in clouds at sub-zero temperatures, 0°C is better defined as the melting point of ice. In this scale a temperature difference of 1 degree Celsius is the same as a 1kelvin increment, but the scale is offset by the temperature at which ice melts (273.15 K).

By international agreement^[29] the Kelvin and Celsius scales are defined by two fixing points: absolute zero and the triple point of Vienna Standard Mean Ocean Water, which is water specially prepared with a specified blend of hydrogen and oxygen isotopes. Absolute zero is defined as precisely 0K and −273.15°C. It is the temperature at which all classical translational motion of the particles comprising matter ceases and they are at complete rest in the classical model. Quantum-mechanically, however, zero-point motion remains and has an associated energy, the zero-point energy. Matter is in its ground state,^[30] and contains no thermal energy. The triple point of water is defined as 273.16K and 0.01°C. This definition serves the following purposes: it fixes the magnitude of the kelvin as being precisely 1 part in 273.16 parts of the difference between absolute zero and the triple point of water; it establishes that one kelvin has precisely the same magnitude as one degree on the Celsius scale; and it establishes the difference between the null points of these scales as being 273.15K (0K = −273.15°C and 273.16K = 0.01°C).

In the United States, the Fahrenheit scale is widely used. On this scale the freezing point of water corresponds to 32 °F and the boiling point to 212 °F. The Rankine scale, still used in fields of chemical engineering in the U.S., is an absolute scale based on the Fahrenheit increment.

Conversion[link]

The following table shows the temperature conversion formulas for conversions to and from the Celsius scale.

Plasma physics[link]

The field of plasma physics deals with phenomena of electromagnetic nature that involve very high temperatures. It is customary to express temperature in electronvolts (eV) or kiloelectronvolts (keV), where 1 eV = 11605K. In the study of QCD matter one routinely encounters temperatures of the order of a few hundred MeV, equivalent to about 10¹²K.

Theoretical foundation[link]

Historically, there are several scientific approaches to the explanation of temperature: the classical thermodynamic description based on macroscopic empirical variables that can be measured in a laboratory; the kinetic theory of gases which relates the macroscopic description to the probability distribution of the energy of motion of gas particles; and a microscopic explanation based on statistical physics and quantum mechanics. In addition, rigorous and purely mathematical treatments have provided an axiomatic approach to classical thermodynamics and temperature.^[31] Statistical physics provides a deeper understanding by describing the atomic behavior of matter, and derives macroscopic properties from statistical averages of microscopic states, including both classical and quantum states. In the fundamental physical description, using natural units, temperature may be measured directly in units of energy. However, in the practical systems of measurement for science, technology, and commerce, such as the modern metric system of units, the macroscopic and the microscopic descriptions are interrelated by the Boltzmann constant, a proportionality factor that scales temperature to the microscopic mean kinetic energy.

The microscopic description in statistical mechanics is based on a model that analyzes a system into its fundamental particles of matter or into a set of classical or quantum-mechanical oscillators and considers the system as a statistical ensemble of microstates. As a collection of classical material particles, temperature is a measure of the mean energy of motion, called kinetic energy, of the particles, whether in solids, liquids, gases, or plasmas. Kinetic energy, a concept of classical mechanics, is one half the product of mass and the square of a particle's velocity. In this mechanical interpretation of thermal motion, the kinetic energies of material particles may reside in the velocity of the particles of their translational or vibrational motion or in the inertia of their rotational modes. In monoatomic perfect gases and, approximately, in most gases, temperature is a measure of the mean particle kinetic energy. It also determines the probability distribution function of the energy. In condensed matter, and particularly in solids, this purely mechanical description is often less useful and the oscillator model provides a better description to account for quantum mechanical phenomena. Temperature determines the statistical occupation of the microstates of the ensemble. The microscopic definition of temperature is only meaningful in the thermodynamic limit, meaning for large ensembles of states or particles, to fulfill the requirements of the statistical model.

In the context of thermodynamics, the kinetic energy is also referred to as thermal energy. The thermal energy may be partitioned into independent components attributed to the degrees of freedom of the particles or to the modes of oscillators in a thermodynamic system. In general, the number of these degrees of freedom that are available for the equipartitioning of energy depend on the temperature, i.e. the energy region of the interactions under consideration. For solids, the thermal energy is associated primarily with the vibrations of its atoms or molecules about their equilibrium position. In an ideal monatomic gas, the kinetic energy is found exclusively in the purely translational motions of the particles. In other systems, vibrational and rotational motions also contribute degrees of freedom.

Kinetic theory of gases[link]

The temperature of an ideal monatomic gas is related to the average kinetic energy of its atoms. In this animation, the size of helium atoms relative to their spacing is shown to scale under 1950 atmospheres of pressure. These atoms have a certain, average speed (slowed down here two trillion fold from room temperature).

The kinetic theory of gases uses the model of the ideal gas to relate temperature to the average kinetic energy of the atoms in a container of gas. Classical mechanics defines the kinetic energy as follows:

Failed to parse (Missing texvc executable; please see math/README to configure.): E_\text{k} = \begin{matrix} \frac 1 2 \end{matrix} mv^2,

where m is the particle mass and v its velocity. The distribution of energies (and thus speeds) of the particles in any gas are given by the Maxwell-Boltzmann distribution. The temperature of a classical ideal gas is related to its average kinetic energy per degree of freedom E_k via the equation:^[32]

Failed to parse (Missing texvc executable; please see math/README to configure.): \overline{E}_\text{k} = \begin{matrix} \frac 1 2 \end{matrix} kT,

where the Boltzmann constant Failed to parse (Missing texvc executable; please see math/README to configure.): k = R/n

(n = Avogadro number, R = ideal gas constant). This relation is valid in the classical regime, i.e. when the particle density is much less than Failed to parse (Missing texvc executable; please see math/README to configure.): 1/\Lambda^{3}

, where Failed to parse (Missing texvc executable; please see math/README to configure.): \Lambda

is the thermal de Broglie wavelength. A monoatomic gas has only the three translational degrees of freedom.

The second law of thermodynamics states that any two given systems when interacting with each other will later reach the same average energy per particle and hence the same temperature.

In a mixture of particles of various masses, the heaviest particles will move slower than lighter particles, but have the same average kinetic energy. A neon atom moves slower relative to a hydrogen molecule of the same kinetic energy; a pollen particle suspended in water moves in a slow Brownian motion among fast moving water molecules.

Zeroth law of thermodynamics[link]

It has long been recognized that if two bodies of different temperatures are brought into thermal connection, conductive or radiative, they exchange heat accompanied by changes of other state variables. Left isolated from other bodies, the two connected bodies eventually reach a state of thermal equilibrium in which no further changes occur. This basic knowledge is relevant to thermodynamics. Some approaches to thermodynamics take this basic knowledge as axiomatic, other approaches select only one narrow aspect of this basic knowledge as axiomatic, and use other axioms to justify and express deductively the remaining aspects of it. The one aspect chosen by the latter approaches is often stated in textbooks as the zeroth law of thermodynamics, but other statements of this basic knowledge are made by various writers.

The usual textbook statement of the zeroth law of thermodynamics is that if two systems are each in thermal equilibrium with a third system, then they are also in thermal equilibrium with each other. This statement is taken to justify a statement that all three systems have the same temperature, but, by itself, it does not justify the idea of temperature as a numerical scale for a concept of hotness which exists on a one-dimensional manifold with a sense of greater hotness. Sometimes the zeroth law is stated to provide the latter justification.^[24] For suitable systems, an empirical temperature scale may be defined by the variation of one of the other state variables, such as pressure, when all other coordinates are fixed. The second law of thermodynamics is used to define an absolute thermodynamic temperature scale for systems in thermal equilibrium.

A temperature scale is based on the properties of some reference system to which other thermometers may be calibrated. One such reference system is a fixed quantity of gas. The ideal gas law indicates that the product of the pressure (p) and volume (V) of a gas is directly proportional to the thermodynamic temperature:^[32]

Failed to parse (Missing texvc executable; please see math/README to configure.): pV = nRT\,\!

where T is temperature, n is the number of moles of gas and R = 8.314472(15) Jmol^-1K^-1 is the gas constant. Reformulating the pressure-volume term as the sum of classical mechanical particle energies in terms of particle mass, m, and root-mean-square particle speed v, the ideal gas law directly provides the relationship between kinetic energy and temperature:^[33]

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle \frac 1 2 mv_{rms}^2 = \frac 3 2 k T.

Thus, one can define a scale for temperature based on the corresponding pressure and volume of the gas: the temperature in kelvins is the pressure in pascals of one mole of gas in a container of one cubic metre, divided by the gas constant. In practice, such a gas thermometer is not very convenient, but other thermometers can be calibrated to this scale.

The pressure, volume, and the number of moles of a substance are all inherently greater than or equal to zero, suggesting that temperature must also be greater than or equal to zero. As a practical matter it is not possible to use a gas thermometer to measure absolute zero temperature since the gasses tend to condense into a liquid long before the temperature reaches zero. It is possible, however, to extrapolate to absolute zero by using the ideal gas law.

Second law of thermodynamics[link]

In the previous section certain properties of temperature were expressed by the zeroth law of thermodynamics. It is also possible to define temperature in terms of the second law of thermodynamics which deals with entropy. Entropy is often thought of as a measure of the disorder in a system. The second law states that any process will result in either no change or a net increase in the entropy of the universe. This can be understood in terms of probability.

For example, in a series of coin tosses, a perfectly ordered system would be one in which either every toss comes up heads or every toss comes up tails. This means that for a perfectly ordered set of coin tosses, there is only one set of toss outcomes possible: the set in which 100% of tosses come up the same. On the other hand, there are multiple combinations that can result in disordered or mixed systems, where some fraction are heads and the rest tails. A disordered system can be 90% heads and 10% tails, or it could be 98% heads and 2% tails, et cetera. As the number of coin tosses increases, the number of possible combinations corresponding to imperfectly ordered systems increases. For a very large number of coin tosses, the combinations to ~50% heads and ~50% tails dominates and obtaining an outcome significantly different from 50/50 becomes extremely unlikely. Thus the system naturally progresses to a state of maximum disorder or entropy.

It has been previously stated that temperature governs the flow of heat between two systems and it was just shown that the universe tends to progress so as to maximize entropy, which is expected of any natural system. Thus, it is expected that there is some relationship between temperature and entropy. To find this relationship, the relationship between heat, work and temperature is first considered. A heat engine is a device for converting thermal energy into mechanical energy, resulting in the performance of work, and analysis of the Carnot heat engine provides the necessary relationships. The work from a heat engine corresponds to the difference between the heat put into the system at the high temperature, q_H and the heat ejected at the low temperature, q_C. The efficiency is the work divided by the heat put into the system or:

Failed to parse (Missing texvc executable; please see math/README to configure.): \textrm{efficiency} = \frac {w_{cy}}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H}

(2)

where w_cy is the work done per cycle. The efficiency depends only on q_C/q_H. Because q_C and q_H correspond to heat transfer at the temperatures T_C and T_H, respectively, q_C/q_H should be some function of these temperatures:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{q_C}{q_H} = f(T_H,T_C)

(3)

Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, a heat engine operating between T₁ and T₃ must have the same efficiency as one consisting of two cycles, one between T₁ and T₂, and the second between T₂ and T₃. This can only be the case if:

Failed to parse (Missing texvc executable; please see math/README to configure.): q_{13} = \frac{q_1 q_2} {q_2 q_3}

which implies:

Failed to parse (Missing texvc executable; please see math/README to configure.): q_{13} = f(T_1,T_3) = f(T_1,T_2)f(T_2,T_3)

Since the first function is independent of T₂, this temperature must cancel on the right side, meaning f(T₁,T₃) is of the form g(T₁)/g(T₃) (i.e. f(T₁,T₃) = f(T₁,T₂)f(T₂,T₃) = g(T₁)/g(T₂)· g(T₂)/g(T₃) = g(T₁)/g(T₃)), where g is a function of a single temperature. A temperature scale can now be chosen with the property that:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{q_C}{q_H} = \frac{T_C}{T_H}

(4)

Substituting Equation 4 back into Equation 2 gives a relationship for the efficiency in terms of temperature:

Failed to parse (Missing texvc executable; please see math/README to configure.): \textrm{efficiency} = 1 - \frac{q_C}{q_H} = 1 - \frac{T_C}{T_H}

(5)

Notice that for T_C = 0 K the efficiency is 100% and that efficiency becomes greater than 100% below 0 K. Since an efficiency greater than 100% violates the first law of thermodynamics, this implies that 0 K is the minimum possible temperature. In fact the lowest temperature ever obtained in a macroscopic system was 20 nK, which was achieved in 1995 at NIST. Subtracting the right hand side of Equation 5 from the middle portion and rearranging gives:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac {q_H}{T_H} - \frac{q_C}{T_C} = 0

where the negative sign indicates heat ejected from the system. This relationship suggests the existence of a state function, S, defined by:

Failed to parse (Missing texvc executable; please see math/README to configure.): dS = \frac {dq_\mathrm{rev}}{T}

(6)

where the subscript indicates a reversible process. The change of this state function around any cycle is zero, as is necessary for any state function. This function corresponds to the entropy of the system, which was described previously. Rearranging Equation 6 gives a new definition for temperature in terms of entropy and heat:

Failed to parse (Missing texvc executable; please see math/README to configure.): T = \frac{dq_\mathrm{rev}}{dS}

(7)

For a system, where entropy S(E) is a function of its energy E, the temperature T is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): {T}^{-1} = \frac{d}{dE} S(E)

(8),

i.e. the reciprocal of the temperature is the rate of increase of entropy with respect to energy.

Definition from statistical mechanics[link]

The previous section elaborated the historical derivation relating entropy and heat. A modern definition of temperature is given by statistical mechanics. It is defined in terms of the fundamental degrees of freedom of a system. Eq.(8) of the previous section is taken to be the defining relation of the temperature. Eq. (7) can be derived from first principles.

Generalized temperature from single particle statistics[link]

It is possible to extend the definition of temperature even to systems of few particles, like in a quantum dot. The generalized temperature is obtained by considering time ensembles instead of configuration space ensembles given in statistical mechanics in the case of thermal and particle exchange between a small system of fermions (N even less than 10) with a single/double occupancy system. The finite quantum grand canonical ensemble,^[34] obtained under the hypothesis of ergodicity and orthodicity, allows to express the generalized temperature from the ratio of the average time of occupation Failed to parse (Missing texvc executable; please see math/README to configure.): \tau ₁ and Failed to parse (Missing texvc executable; please see math/README to configure.): \tau ₂ of the single/double occupancy system:^[35]

Failed to parse (Missing texvc executable; please see math/README to configure.): T = k^{-1} \ln 2\frac{\tau_\mathrm{2}}{\tau_\mathrm{1}} \left(E - E_{F} \left(1+\frac{3}{2N}\right) \right),

where E_F is the Fermi energy which tends to the ordinary temperature when N goes to infinity.

Negative temperature[link]

On the empirical temperature scales, which are not referenced to absolute zero, a negative temperature is one below the zero-point of the scale used. For example, dry ice has a sublimation temperature of −78.5°C which is equivalent to −109.3°F. On the absolute Kelvin scale, however, this temperature is 194.6 K. On the absolute scale of thermodynamic temperature no material can exhibit a temperature smaller than or equal to 0 K, both of which are forbidden by the third law of thermodynamics.

In the quantum mechanical description of electron and nuclear spin systems that have a limited number of possible states, and therefore a discrete upper limit of energy they can attain, it is possible to obtain a negative temperature, which is numerically indeed less than absolute zero. However, this is not the macroscopic temperature of the material, but instead the temperature of only very specific degrees of freedom, that are isolated from others and do not exchange energy by virtue of the equipartition theorem.

A negative temperature is experimentally achieved with suitable radio frequency techniques that cause a population inversion of spin states from the ground state. As the energy in the system increases upon population of the upper states, the entropy increases as well, as the system becomes less ordered, but attains a maximum value when the spins are evenly distributed among ground and excited states, after which it begins to decrease, once again achieving a state of higher order as the upper states begin to fill exclusively. At the point of maximum entropy, the temperature function shows the behavior of a singularity, because the slope of the entropy function decreases to zero at first and then turns negative. Since temperature is the inverse of the derivative of the entropy, the temperature formally goes to infinity at this point, and switches to negative infinity as the slope turns negative. At energies higher than this point, the spin degree of freedom therefore exhibits formally a negative thermodynamic temperature. As the energy increases further by continued population of the excited state, the negative temperature approaches zero asymptotically.^[36] As the energy of the system increases in the population inversion, a system with a negative temperature is not colder than absolute zero, but rather it has a higher energy than at positive temperature, and may be said to be in fact hotter at negative temperatures. When brought into contact with a system at a positive temperature, energy will be transferred from the negative temperature regime to the positive temperature region.

Examples of temperature[link]

• ^A For Vienna Standard Mean Ocean Water at one standard atmosphere (101.325 kPa) when calibrated strictly per the two-point definition of thermodynamic temperature.
• ^B The 2500 K value is approximate. The 273.15 K difference between K and °C is rounded to 300 K to avoid false precision in the Celsius value.
• ^C For a true black-body (which tungsten filaments are not). Tungsten filaments' emissivity is greater at shorter wavelengths, which makes them appear whiter.
• ^D Effective photosphere temperature. The 273.15 K difference between K and °C is rounded to 273 K to avoid false precision in the Celsius value.
• ^E The 273.15 K difference between K and °C is without the precision of these values.
• ^F For a true black-body (which the plasma was not). The Z machine's dominant emission originated from 40 MK electrons (soft x–ray emissions) within the plasma.

See also[link]

Notes[link]

References[link]

Further reading[link]

• Chang, Hasok (2004). Inventing Temperature: Measurement and Scientific Progress. Oxford: Oxford University Press. ISBN 978-0-19-517127-3.
• Zemansky, Mark Waldo (1964). Temperatures Very Low and Very High. Princeton, N.J.: Van Nostrand.
• T. J. Quinn (1983), Temperature, Academic Press, London.

External links[link]

Wikimedia Commons has media related to: Temperature

Look up temperature in Wiktionary, the free dictionary.

• An elementary introduction to temperature aimed at a middle school audience
• What is Temperature? An introductory discussion of temperature as a manifestation of kinetic theory.
• from Oklahoma State University

Sean Paul
Sean Paul at the International Reggae and World Music Awards.
Background information
Birth name	Sean Paul Ryan Francis Henriques
Born	(1973-01-09) January 9, 1973 (age 39) Kingston, Jamaica
Genres	Dancehall, reggae, R&B, ragga
Occupations	Musician, singer-songwriter, producer, actor
Years active	1996–present
Labels	VP, Atlantic
Website	allseanpaul.com

Sean Paul Ryan Francis Henriques^[1] (born January 9, 1973),^[1][2] who performs under stage name Sean Paul, is a Jamaican dancehall and reggae artist.

Life and career[link]

1973–96: Early life[link]

Sean Paul was born in Kingston and spent his early years in Upper Andrew Parish, a few miles north of Kingston.^[2] His parents, Garth and Frances, were both talented athletes, and his mother is a well-known painter.^[3] His paternal grandfather was a Sephardic Jew whose family emigrated from Portugal, and his paternal grandmother was Afro-Caribbean; his mother is of English and Chinese Jamaican descent.^[4][5] Sean Paul was raised as a Catholic.^[6] Many members of his family are swimmers. His grandfather was on the first Jamaican men's national water polo team. His father also played water polo for the team in the 1960s, and competed in long-distance swimming, while Sean Paul's mother was a backstroke swimmer. Sean Paul played for the national water polo team from the age of thirteen to twenty-one, when he gave up the sport in order to launch his musical career. He attended the Wolmers High School for Boys, Belair School, Hillel Academy High School, and the College of Arts, Science, and Technology, now known as the University of Technology, where he was trained in commerce with a view to pursuing an occupation in hotel management.

[edit] 1998–2000: Stage One

Sean Paul's manager and producer Jeremy Harding first heard about the singer when his brother told him about seeing someone at a small open mic event in Kingston who sounded a lot like the popular dancehall DJ and toaster Super Cat.^[7] Harding eventually met the singer when Sean Paul came by his studio to ask for some advice. During the meeting Paul recorded a vocal over Harding's rhythm track and in the process created the song "Baby Girl".^[7] Sean Paul began hanging out at the studio every day, and the pair collaborated on several more tracks. When they recorded "Infiltrate" they decided they had something good enough to get on the radio. As Sean Paul started to attract local attention, Harding began looking after his affairs. He later told HitQuarters that his support of Paul's fledgling career initially led him assuming the roles of "DJ, manager, road manager and security guard."^[7] Sean Paul made a quick cameo appearance in the 1998 film Belly on stage performing. He made a very successful collaboration with DMX & Mr. Vegas (Top Shotter) as a soundtrack of the film. In 2000, Sean Paul released his debut album, Stage One with VP Records.

[edit] 2001–02: Dutty Rock

In 2002, he began working extensively with a team of producers and choreographers from Toronto, namely Jae Blaze and Blaze Entertainment and announced the release of his second album, Dutty Rock. Pushed by the success of the singles "Gimme the Light" and the Billboard Hot 100 topper, "Get Busy", the album was a worldwide success, eventually selling over six million copies. Simultaneously, Sean Paul was heard on Beyoncé's U.S. #1 single "Baby Boy" and Blu Cantrell's "Breathe", a chart hit in Europe. Both helped to push his reputation further still in the United States. He appeared on Punk'd, 106 & Park, Sean Paul Respect, Making the Video ("Get Busy", "Gimme the Light", and "Like Glue") and his music videos have been broadcast on MTV and BET. Paul's biggest hits included "Get Busy", "Like Glue", "Gimme the Light", "Baby Boy", and "I'm Still in Love with You".

[edit] 2003–05: The Trinity

Sean Paul in September 2005.

Sean Paul's third album The Trinity was released in on September 27, 2005. The album produced five big hits, "We Be Burnin'", "Ever Blazin'", "Give It Up to Me", "Never Gonna Be The Same" and the U.S. chart-topping smash hit "Temperature". The video of "Give It Up to Me" (featuring Keyshia Cole) was also featured in the movie Step Up in 2006. He was nominated for four awards at the 2006 Billboard Music Awards, including male artist of the year, rap artist of the year, hot 100 single of the year, and pop single of the year for his hit "Temperature".^[8] He also won an American Music Award for "(When You Gonna) Give It Up To Me" beating Kanye West and Nick Lachey who were also nominated for the award. His song "Send It On" from "The Trinity" featured on the 2005 Vauxhall Corsa advert. Sean Paul often contributes his songs to various Riddim Driven albums (by VP Records). In March 2007, he returned to his native Jamaica to perform at the Cricket World Cup 2007 opening ceremony. Sean Paul appears on the game Def Jam: Fight for NY as part of Snoop Dogg's crew and again in the game's sequel, Def Jam Icon.

[edit] 2006–10: Imperial Blaze

The newest Sean Paul album entitled "Imperial Blaze" was released on August 18, 2009. The lead single, “So Fine”, which was produced by Stephen “Di Genius” McGregor, premiered on Sean Paul's official website on April 26, 2009.^{[9][dead link][10]} Speaking to Pete Lewis of 'Blues & Soul' magazine in August 2009, Sean Paul stated that 'Imperial Blaze' "Actually signifies 'The King's Fire'. It's that thing inside of you that gives you the desire to do whatever you do, and be the best in the world at it."^[11] The new album consists of 20 tracks including "So Fine", "Press it Up", "She Want Me", "Private Party" which are party tracks and also love songs such as "Hold My Hand" (feat Keri Hilson), "Lately", "Now That I've Got Your Love" among others. Producers on the album include Don Corleone, Jeremy Harding, and Sean's brother Jason 'Jigzagula' Henriques.^[11] All the full songs of the album have been added to Sean Paul's Myspace page^[12] on the day of release of the album.

Up until now there have been eight music videos: "Always On My Mind (with Da'Ville)", "Give It To You (with Eve)", "Watch Them Roll", "Back It Up" (with Left Side/Mr. Evil), "(I Wanna See You) Push It Baby" (with Pretty Ricky), "Hit 'Em" (with Fahrenheit and his brother Jason "Jigzagula" Henriques), "Come Over" with Estelle,^[13] and also the video of his first single, "So Fine" from the new album. He has recently been featured in Shaggy's video, "Save A Life", which also includes appearances from Elephant Man and Da'Ville, among others. In an effort to raise money for a children's hospital, Shaggy, Sean Paul and others will be having a benefit concert. All proceeds will go towards getting new equipment and technology 'For Aid to the Bustamante Hospital for Children'. In an interview in 2009 he says he is planning to release a new album in 2011. During the premiere for MNET's Big Brother Africa 5: All-Stars on July 18, 2010, he performed his songs "Temperature", "Hold My Hand", and "So Fine".

[edit] 2011–present: Tomahawk Technique

The first single Got 2 Luv U features vocals from American singer Alexis Jordan. It was released on the July 19, 2011 by Atlantic Records. The song was written by Sean Paul, Ryan Tedder and Stargate, and it was produced by Stargate.^[14][15][16]

She Doesn't Mind is the second single from the album. It was written by Sean Paul, Shellback and Benny Blanco and was produced by Shellback and Benny Blanco. It was released on the September 29, 2011 on NRJ & Skyrock (French radios), and to iTunes on October 31.^[17] Like its proceeder, "Got 2 Luv U", featuring Alexis Jordan, it topped the charts in Switzerland, but it debuted at that spot. Sean appeared on the Never Mind the Buzzcocks episode which aired on November 21, 2011.

Sean Paul is featured in the music video for the Simple Plan song Summer Paradise.

Discography[link]

Studio albums

• Stage One (2000)
• Dutty Rock (2002)
• The Trinity (2005)
• Imperial Blaze (2009)
• Tomahawk Technique (2012)

Mixtapes

• The Odyssey Mixtape (2009)

Filmography[link]

Les Nubians is an R&B Grammy-nominated^[1][2] duo composed of sisters Hélène and Célia Faussart from Paris, France. In 1985 the sisters moved with their parents to Chad. Seven years later, they returned to Bordeaux, France, and began singing a cappella, producing poetry slams in Bordeaux and Paris, and singing background vocals for various artists worldwide. The duo's debut album Princesses Nubiennes was released by Virgin Records, France, in 1998.

They have become one of the most successful French-language musical groups in the U.S., best known for their Billboard R&B Single "Makeda"^[3] from their Grammy nominated album Princesses Nubiennes.^[1] Les Nubians were the 1999 Soul Train Lady of Soul Awards winners for Best New Artist, Group or Duo and received two NAACP Image Awards nominations in 2000.^[1]

The duo executive-produced Echos, Chapter One in 2005 on their label Nubiatik; a joint venture with Triloka Records. The project features artists from France, and the U.S. performing poetry and music from the urban edge. Echos, Chapter One: Nubian Voyager was released in 2006 as a book accompaniment to the CD.

Discography[link]

Albums[link]

• Princesses Nubiennes (1998) Virgin Records
• One Step Forward (2003) Virgin Records, Higher Octave
• Echos, Chapter One (2005) Nubiatik
• Nü Revolution (2011) Nubiatik, Shanachie

Soundtracks[link]

• 2011: "Beat The World"

References[link]

External links[link]

• Official Site

For the radio personality, see Little Walter DeVenne.

Little Walter, born Marion Walter Jacobs (May 1, 1930 – February 15, 1968), was an American blues harmonica player, whose revolutionary approach to his instrument has earned him comparisons to Charlie Parker and Jimi Hendrix,^[1] for innovation and impact on succeeding generations. His virtuosity and musical innovations fundamentally altered many listeners' expectations of what was possible on blues harmonica.^[2] Little Walter was inducted to the The Rock and Roll Hall of Fame in 2008 in the "sideman" category^[3][4] making him the first and only artist ever to be inducted specifically for his work as a harmonica player.

Biography[link]

Early years[link]

Jacobs was born in Marksville, Louisiana and raised in Rapides Parish, Louisiana, where he first learned to play the harmonica. After quitting school by the age of 12, Jacobs left rural Louisiana and travelled around working odd jobs and busking on the streets of New Orleans, Memphis, Helena, Arkansas and St. Louis. He honed his musical skills on harmonica and guitar with Sonny Boy Williamson II, Sunnyland Slim, Honeyboy Edwards and others.

Arriving in Chicago in 1945, he occasionally found work as a guitarist but garnered more attention for his already highly developed harmonica work. According to fellow Chicago bluesman Floyd Jones, Little Walter's first recording was an unreleased demo recorded soon after he arrived in Chicago on which Walter played guitar backing Jones.^[5] Jacobs reportedly grew frustrated with having his harmonica drowned out by electric guitarists, and adopted a simple, but previously little-used method: He cupped a small microphone in his hands along with his harmonica, and plugged the microphone into a public address or guitar amplifier. He could thus compete with any guitarist's volume. However, unlike other contemporary blues harp players such as Sonny Boy Williamson I and Snooky Pryor, who had also begun using the then-new technology around the same time solely for added volume, Little Walter purposely pushed his amplifiers beyond their intended technical limitations, using the amplification to explore and develop radical new timbres and sonic effects previously unheard from a harmonica, or any other instrument.^[1] Madison Deniro wrote a small biographical piece on Little Walter stating that "He was the first musician of any kind to purposely use electronic distortion."^[6]

Success[link]

Jacobs made his first released recordings in 1947 for Bernard Abrams' tiny Ora-Nelle label, which operated out of the back room of Abrams' Maxwell Radio and Records store in the heart of the Maxwell Street market area in Chicago.^[7] These and several other early Little Walter recordings, like many blues harp recordings of the era, owed a strong stylistic debt to pioneering blues harmonica player Sonny Boy Williamson I (John Lee Williamson). Little Walter joined Muddy Waters' band in 1948, and by 1950, he was playing acoustic (unamplified) harmonica on Muddy's recordings for Chess Records; the first appearance on record of Little Walter's amplified harmonica sound was on Muddy's "Country Boy"/"Too Young To Know" (Chess 1452), recorded on July 11, 1951. For years after his departure from Muddy's band in 1952, Little Walter continued to be brought in to play on his recording sessions, and as a result his harmonica is featured on most of Muddy's classic recordings from the 1950s.^[8] As a guitarist, Little Walter recorded three songs for the small Parkway label with Muddy Waters and Baby Face Leroy Foster (reissued on CD as "The Blues World of Little Walter" from Delmark Records in 1993), as well as on a session for Chess backing pianist Eddie Ware; his guitar work was also featured occasionally on early Chess sessions with Muddy Waters and Jimmy Rogers.

Jacobs had put his career as a bandleader on hold when he joined Muddy's band, but stepped back out front once and for all when he recorded as a bandleader for Chess's subsidiary label Checker Records on 12 May 1952; the first completed take of the first song attempted at his debut session became his first hit, spending eight weeks in the #1 position on the Billboard magazine R&B charts – the song was "Juke", and it is still the only harmonica instrumental ever to become a #1 hit on the R&B charts. (Three other harmonica instrumentals by Little Walter also reached the Billboard R&B top 10: "Off the Wall" reached #8, "Roller Coaster" achieved #6, and "Sad Hours" reached the #2 position while Juke was still on the charts.) "Juke" was the biggest hit to date for Chess and its affiliated labels, and one of the biggest national R&B hits of 1952, securing Walter's position on the Chess artist roster for the next decade. Little Walter scored fourteen top-ten hits on the Billboard R&B charts between 1952 and 1958, including two #1 hits (the second being "My Babe" in 1955), a level of commercial success never achieved by his former boss Waters, nor by his fellow Chess blues artists Howlin' Wolf and Sonny Boy Williamson II. Following the pattern of "Juke", most of Little Walter's single releases in the 1950s featured a vocal performance on one side, and an instrumental on the other. Many of Walter's vocal numbers were originals which he or Chess A&R man Willie Dixon wrote or adapted and updated from earlier blues themes. In general, his sound was more modern and uptempo than the popular Chicago blues of the day, with a jazzier conception and less rhythmically rigid approach than other contemporary blues harmonica players.^[1]

Jacobs was frequently utilized on records as a harmonica accompanist behind others in the Chess stable of artists, including Jimmy Rogers, John Brim, Rocky Fuller, Memphis Minnie, The Coronets, Johnny Shines, Floyd Jones, Bo Diddley, and Shel Silverstein, and on other record labels backing Otis Rush, Johnny Young, and Robert Nighthawk.

Jacobs suffered from alcoholism and had a notoriously short temper which led to a decline in his fame and fortunes beginning in the late 1950s, although he did tour Europe twice, in 1964 and 1967. (The long-circulated story that he toured the United Kingdom with The Rolling Stones in 1964 has since been refuted by Keith Richards). The 1967 European tour, as part of the American Folk Blues Festival, resulted in the only film/video footage of Little Walter performing that is known to exist. Footage of Little Walter backing Hound Dog Taylor and Koko Taylor on a television program in Copenhagen, Denmark on 11 October 1967 was released on DVD in 2004. Further video of another recently discovered TV appearance in Germany during this tour, showing Little Walter performing his songs "My Babe", "Mean Old World", and others were released on DVD in Europe in January 2009, and is the only known footage of Little Walter singing; other TV appearances in the UK and the Netherlands have been documented, but no footage of these has been uncovered. Jacobs recorded and toured only infrequently in the 1960s, playing mainly in and around Chicago.

In 1967 Chess released a studio album featuring Little Walter with Bo Diddley and Muddy Waters titled Super Blues.

Death[link]

A few months after returning from his second European tour, he was involved in a fight while taking a break from a performance at a nightclub on the South Side of Chicago. The relatively minor injuries sustained in this altercation aggravated and compounded damage he had suffered in previous violent encounters, and he died in his sleep at the apartment of a girlfriend at 209 E. 54th St. in Chicago early the following morning.^[1][9] The official cause of death indicated on his death certificate was "coronary thrombosis" (a blood clot in the heart); evidence of external injuries was so insignificant that police reported that his death was of "unknown or natural causes",^[9] and there were no external injuries noted on the death certificate.^[1] His body was buried at St. Mary's Cemetery in Evergreen Park, IL on February 22, 1968.^[9]

Legacy[link]

Music journalist Bill Dahl described Little Walter as "king of all post-war blues harpists", who "took the humble mouth organ in dazzling amplified directions that were unimaginable prior to his ascendancy."^[2] His legacy has been enormous: he is widely credited by blues historians as the artist primarily responsible for establishing the standard vocabulary for modern blues and blues rock harmonica players.^[2][1] His influence can be heard in varying degrees in virtually every modern blues harp player who came along in his wake, from blues greats such as Junior Wells, James Cotton, George "Harmonica" Smith, Carey Bell, and Big Walter Horton, through modern-day masters Sugar Blue, Billy Branch, Kim Wilson, Rod Piazza, William Clarke, and Charlie Musselwhite, in addition to blues-rock crossover artists such as Paul Butterfield and John Popper of the band Blues Traveler. ^[1] Little Walter was portrayed in the 2008 film, Cadillac Records, by Columbus Short.

Little Walter's daughter, Marion Diaz Reacco has established the Little Walter Foundation in Chicago, to preserve the legacy and genius of Little Walter. The foundation aims to create programs for the creative arts, including music, animation and video.

Awards and recognition[link]

• 1986 – Blues Hall of Fame: "Juke" (Classics of Blues Recordings – Singles or Album Tracks category)^[10]
• 1991 – Blues Hall of Fame: Best of Little Walter (Classics of Blues Recordings – Albums category)^[10]
• 1995 – Rock and Roll Hall of Fame: "Juke" (500 Songs that Shaped Rock and Roll)^[11]
• 2003 – Rolling Stone: Best of Little Walter (#198 on list of The 500 Greatest Albums of All Time)^[12]
• 2008 – Grammy Awards: "Juke" (Grammy Hall of Fame Award)^[13]
• 2008 – Rock and Roll Hall of Fame: Little Walter inducted (Sideman category)
• 2008 – Blues Hall of Fame: "My Babe" (Classics of Blues Recordings – Singles or Album Tracks category)^[10]
• 2009 – Grammy Awards The Complete Chess Masters: 1950–1967 (Best Historical Album Winner)

Discography[link]

Charting singles[link]

Little Walter released fifteen singles that made the charts during his career. These were issued on Checker, a Chess subsidiary; the chart information is the peak position the single reached on the Billboard R&B chart.