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Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied on various reference points and thermometric substances for definition. The most common scales are the Celsius scale with the unit symbol °C (formerly called centigrade), the Fahrenheit scale (°F), and the Kelvin scale (K), the latter being used predominantly for scientific purposes. The kelvin is one of the seven base units in the International System of Units (SI).
Absolute zero, i.e. zero kelvin or −273.15 °C, is the lowest point in the thermodynamic temperature scale. Experimentally, it can be approached only very closely, but not actually reached, as recognized in the third law of thermodynamics. It would be impossible to extract energy as heat from a body at that temperature.
Temperature is important in all fields of natural science, including physics, chemistry, Earth science, astronomy, medicine, biology, ecology, material science, metallurgy, mechanical engineering and geography as well as most aspects of daily life.
The Celsius scale (°C) is used for common temperature measurements in most of the world. It is an empirical scale that developed historically, which led to its zero point being defined as the freezing point of water, and as the boiling point of water, both at atmospheric pressure at sea level. Because of the 100degree interval, it was called a centigrade scale.Middleton, W.E.K. (1966), pp. 89–105. Since the standardization of the kelvin in the International System of Units, it has subsequently been redefined in terms of the equivalent fixing points on the Kelvin scale, and so that a temperature increment of one degree Celsius is the same as an increment of one kelvin, though numerically the scales differ by an offset of exactly 273.15.
The Fahrenheit scale is in common use in the United States. Water freezes at and boils at at sealevel atmospheric pressure.
Besides the internationally agreed Kelvin scale, there is also a thermodynamic temperature scale, invented by Lord Kelvin, also with its numerical zero at the absolute zero of temperature, but directly relating to purely macroscopic thermodynamics concepts, including the macroscopic entropy, though microscopically referable to the Gibbs statistical mechanical definition of entropy for the canonical ensemble, that takes interparticle potential energy into account, as well as independent particle motion so that it can account for measurements of temperatures near absolute zero. This scale has a reference temperature at the triple point of water, the numerical value of which is defined by measurements using the aforementioned internationally agreed Kelvin scale.
Historically, the temperature of the triple point of water was defined as exactly 273.16 K. Today it is an empirically measured quantity. The freezing point of water at sealevel atmospheric pressure occurs at very close to ().
In spite of these limitations, most generally used practical thermometers are of the empirically based kind. Especially, it was used for calorimetry, which contributed greatly to the discovery of thermodynamics. Nevertheless, empirical thermometry has serious drawbacks when judged as a basis for theoretical physics. Empirically based thermometers, beyond their base as simple direct measurements of ordinary physical properties of thermometric materials, can be recalibrated, by use of theoretical physical reasoning, and this can extend their range of adequacy.
Apart from the absolute zero of temperature, the Kelvin temperature of a body in a state of internal thermodynamic equilibrium is defined by measurements of suitably chosen of its physical properties, such as have precisely known theoretical explanations in terms of the Boltzmann constant. That constant refers to chosen kinds of motion of microscopic particles in the constitution of the body. In those kinds of motion, the particles move individually, without mutual interaction. Such motions are typically interrupted by interparticle collisions, but for temperature measurement, the motions are chosen so that, between collisions, the noninteractive segments of their trajectories are known to be accessible to accurate measurement. For this purpose, interparticle potential energy is disregarded.
In an ideal gas, and in other theoretically understood bodies, the Kelvin temperature is defined to be proportional to the average kinetic energy of noninteractively moving microscopic particles, which can be measured by suitable techniques. The proportionality constant is a simple multiple of the Boltzmann constant. If molecules, atoms, or electrons,Germer, L.H. (1925). 'The distribution of initial velocities among thermionic electrons', Phys. Rev., 25: 795–807. hereTurvey, K. (1990). 'Test of validity of Maxwellian statistics for electrons thermionically emitted from an oxide cathode', European Journal of Physics, 11(1): 51–59. here are emitted from material and their velocities are measured, the spectrum of their velocities often nearly obeys a theoretical law called the Maxwell–Boltzmann distribution, which gives a wellfounded measurement of temperatures for which the law holds.Zeppenfeld, M., Englert, B.G.U., Glöckner, R., Prehn, A., Mielenz, M., Sommer, C., van Buuren, L.D., Motsch, M., Rempe, G. (2012). There have not yet been successful experiments of this same kind that directly use the Fermi–Dirac distribution for thermometry, but perhaps that will be achieved in the future.Miller, J. (2013).
The speed of sound in a gas can be calculated theoretically from the Molecule character of the gas, from its temperature and pressure, and from the value of the Boltzmann constant. For a gas of known molecular character and pressure, this provides a relation between temperature and the Boltzmann constant. Those quantities can be known or measured more precisely than can the thermodynamic variables that define the state of a sample of water at its triple point. Consequently, taking the value of the Boltzmann constant as a primarily defined reference of exactly defined value, a measurement of the speed of sound can provide a more precise measurement of the temperature of the gas.de Podesta, M., Underwood, R., Sutton, G., Morantz, P, Harris, P, Mark, D.F., Stuart, F.M., Vargha, G., Machin, M. (2013). A lowuncertainty measurement of the Boltzmann constant, Metrologia, 50 (4): S213–S216, BIPM & IOP Publishing Ltd
Measurement of the spectrum of electromagnetic radiation from an ideal threedimensional black body can provide an accurate temperature measurement because the frequency of maximum spectral radiance of blackbody radiation is directly proportional to the temperature of the black body; this is known as Wien's displacement law and has a theoretical explanation in Planck's law and the Bose–Einstein law.
Measurement of the spectrum of noisepower produced by an electrical resistor can also provide accurate temperature measurement. The resistor has two terminals and is in effect a onedimensional body. The BoseEinstein law for this case indicates that the noisepower is directly proportional to the temperature of the resistor and to the value of its resistance and to the noise bandwidth. In a given frequency band, the noisepower has equal contributions from every frequency and is called Johnson noise. If the value of the resistance is known then the temperature can be found.Quinn, T.J. (1983), pp. 98–107.Schooley, J.F. (1986), pp. 138–143.
Kinetic theory provides a microscopic account of temperature for some bodies of material, especially gases, based on macroscopic systems' being composed of many microscopic particles, such as molecules and of various species, the particles of a species being all alike. It explains macroscopic phenomena through the classical mechanics of the microscopic particles. The equipartition theorem of kinetic theory asserts that each classical degree of freedom of a freely moving particle has an average kinetic energy of where denotes the Boltzmann constant. The translational motion of the particle has three degrees of freedom, so that, except at very low temperatures where quantum effects predominate, the average translational kinetic energy of a freely moving particle in a system with temperature will be .
Molecules, such as oxygen (O_{2}), have more degrees of freedom than single spherical atoms: they undergo rotational and vibrational motions as well as translations. Heating results in an increase of temperature due to an increase in the average translational kinetic energy of the molecules. Heating will also cause, through , the energy associated with vibrational and rotational modes to increase. Thus a diatomic gas will require more energy input to increase its temperature by a certain amount, i.e. it will have a greater heat capacity than a monatomic gas.
As noted above, the speed of sound in a gas can be calculated from the molecular character of the gas, from its temperature and pressure, and from the value of the Boltzmann constant. Taking the value of the Boltzmann constant as a primarily defined reference of exactly defined value, a measurement of the speed of sound can provide a more precise measurement of the temperature of the gas.
It is possible to measure the average kinetic energy of constituent microscopic particles if they are allowed to escape from the bulk of the system, through a small hole in the containing wall. The spectrum of velocities has to be measured, and the average calculated from that. It is not necessarily the case that the particles that escape and are measured have the same velocity distribution as the particles that remain in the bulk of the system, but sometimes a good sample is possible.
The thermodynamic temperature is said to be absolute for two reasons. One is that its formal character is independent of the properties of particular materials. The other reason is that its zero is, in a sense, absolute, in that it indicates absence of microscopic classical motion of the constituent particles of matter, so that they have a limiting specific heat of zero for zero temperature, according to the third law of thermodynamics. Nevertheless, a thermodynamic temperature does in fact have a definite numerical value that has been arbitrarily chosen by tradition and is dependent on the property of particular materials; it is simply less arbitrary than relative "degrees" scales such as Celsius scale and Fahrenheit scale. Being an absolute scale with one fixed point (zero), there is only one degree of freedom left to arbitrary choice, rather than two as in relative scales. For the Kelvin scale since May 2019, by international convention, the choice has been made to use knowledge of modes of operation of various thermometric devices, relying on microscopic kinetic theories about molecular motion. The numerical scale is settled by a conventional definition of the value of the Boltzmann constant, which relates macroscopic temperature to average microscopic kinetic energy of particles such as molecules. Its numerical value is arbitrary, and an alternate, less widely used absolute temperature scale exists called the Rankine scale, made to be aligned with the Fahrenheit scale as Kelvin scale is with Celsius.
The thermodynamic definition of temperature is due to Kelvin. It is framed in terms of an idealized device called a Carnot engine, imagined to run in a fictive continuous Carnot cycle that traverse a cycle of states of its working body. The engine takes in a quantity of heat from a hot reservoir and passes out a lesser quantity of waste heat to a cold reservoir. The net heat energy absorbed by the working body is passed, as thermodynamic work, to a work reservoir, and is considered to be the output of the engine. The cycle is imagined to run so slowly that at each point of the cycle the working body is in a state of thermodynamic equilibrium. The successive processes of the cycle are thus imagined to run reversibly with no entropy production. Then the quantity of entropy taken in from the hot reservoir when the working body is heated is equal to that passed to the cold reservoir when the working body is cooled. Then the absolute or thermodynamic temperatures, and , of the reservoirs are defined such that.
The zeroth law of thermodynamics allows this definition to be used to measure the absolute or thermodynamic temperature of an arbitrary body of interest, by making the other heat reservoir have the same temperature as the body of interest.
Kelvin's original work postulating absolute temperature was published in 1848. It was based on the work of Carnot, before the formulation of the first law of thermodynamics. Carnot had no sound understanding of heat and no specific concept of entropy. He wrote of 'caloric' and said that all the caloric that passed from the hot reservoir was passed into the cold reservoir. Kelvin wrote in his 1848 paper that his scale was absolute in the sense that it was defined "independently of the properties of any particular kind of matter". His definitive publication, which sets out the definition just stated, was printed in 1853, a paper read in 1851.Thomson, W. (Lord Kelvin) (1848).Thomson, W. (Lord Kelvin) (1851).Partington, J.R. (1949), pp. 175–177.Roberts, J.K., Miller, A.R. (1928/1960), pp. 321–322.
Numerical details were formerly settled by making one of the heat reservoirs a cell at the triple point of water, which was defined to have an absolute temperature of 273.16 K.Quinn, T.J. (1983). Temperature, Academic Press, London, , pp. 160–162. Nowadays, the numerical value is instead obtained from measurement through the microscopic statistical mechanical international definition, as above.
In particular, when the body is described by stating its internal energy , an extensive variable, as a function of its entropy , also an extensive variable, and other state variables , with ), then the temperature is equal to the partial derivative of the internal energy with respect to the entropy:Herbert Callen (1960/1985), Thermodynamics and an Introduction to Thermostatistics, (first edition 1960), second edition 1985, John Wiley & Sons, New York, , pp. 146–148.
Likewise, when the body is described by stating its entropy as a function of its internal energy , and other state variables , with , then the reciprocal of the temperature is equal to the partial derivative of the entropy with respect to the internal energy:Kondepudi, D., Ilya Prigogine (1998). Modern Thermodynamics. From Heat Engines to Dissipative Structures, John Wiley, Chichester, , pp. 115–116.
The above definition, equation (1), of the absolute temperature, is due to Kelvin. It refers to systems closed to the transfer of matter and has a special emphasis on directly experimental procedures. A presentation of thermodynamics by Gibbs starts at a more abstract level and deals with systems open to the transfer of matter; in this development of thermodynamics, the equations (2) and (3) above are actually alternative definitions of temperature.Tisza, L. (1966). Generalized Thermodynamics, M.I.T. Press, Cambridge MA, p. 58.
It makes good sense, for example, to say of the extensive variable , or of the extensive variable , that it has a density per unit volume or a quantity per unit mass of the system, but it makes no sense to speak of the density of temperature per unit volume or quantity of temperature per unit mass of the system. On the other hand, it makes no sense to speak of the internal energy at a point, while when local thermodynamic equilibrium prevails, it makes good sense to speak of the temperature at a point. Consequently, the temperature can vary from point to point in a medium that is not in global thermodynamic equilibrium, but in which there is local thermodynamic equilibrium.
Thus, when local thermodynamic equilibrium prevails in a body, the temperature can be regarded as a spatially varying local property in that body, and this is because the temperature is an intensive variable.
When two systems in thermal contact are at the same temperature no heat transfers between them. When a temperature difference does exist heat flows spontaneously from the warmer system to the colder system until they are in thermal equilibrium. Such heat transfer occurs by conduction or by thermal radiation.Maxwell, J.C. (1872). Theory of Heat, third edition, Longmans, Green, London, p. 32.Tait, P.G. (1884). Heat, Macmillan, London, Chapter VII, pp. 39–40.Planck, M. (1897/1903). Treatise on Thermodynamics, translated by A. Ogg, Longmans, Green, London, pp. 1–2.Planck, M. (1914), The Theory of Heat Radiation , second edition, translated into English by M. Masius, Blakiston's Son & Co., Philadelphia, reprinted by Kessinger.
Experimental physicists, for example Galileo and Newton,Tait, P.G. (1884). Heat, Macmillan, London, Chapter VII, pp. 42, 103–117. found that there are indefinitely many empirical temperature scales. Nevertheless, the zeroth law of thermodynamics says that they all measure the same quality. This means that for a body in its own state of internal thermodynamic equilibrium, every correctly calibrated thermometer, of whatever kind, that measures the temperature of the body, records one and the same temperature. For a body that is not in its own state of internal thermodynamic equilibrium, different thermometers can record different temperatures, depending respectively on the mechanisms of operation of the thermometers.
Except for a system undergoing a order parameter phase transition such as the melting of ice, as a closed system receives heat, without a change in its volume and without a 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 a change in external force fields acting on it, decreases its temperature.Truesdell, C., Bharatha, S. (1977). The Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines, Rigorously Constructed upon the Foundation Laid by S. Carnot and F. Reech, Springer, New York, , p. 20.
For example, if the change is an increase in temperature at constant volume, with no phase change and no chemical change, then the temperature of the body rises and its pressure increases. The quantity of heat transferred, , divided by the observed temperature change, , is the body's heat capacity at constant volume:
Temperature is measured with thermometers that may be calibration 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 scientists measure temperature using the Celsius scale and thermodynamic temperature using the Kelvin scale, which is the Celsius scale offset so that its null point is = , or absolute zero. Many engineering fields in the US, notably hightech and US federal specifications (civil and military), also use the Kelvin and Celsius scales. Other engineering fields in the US also rely upon the Rankine scale (a shifted Fahrenheit scale) when working in thermodynamicrelated disciplines such as combustion.
For everyday applications, it is often convenient to use the Celsius scale, in which corresponds very closely to the freezing point of water and is its boiling point at sea level. Because liquid droplets commonly exist in clouds at subzero temperatures, is better defined as the melting point of ice. In this scale, a temperature difference of 1 degree Celsius is the same as a increment, but the scale is offset by the temperature at which ice melts ().
By international agreement, The kelvin in the SI Brochure until May 2019, the Kelvin and Celsius scales were 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 was defined as precisely and . 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. Quantummechanically, however, zeropoint motion remains and has an associated energy, the zeropoint energy. Matter is in its ground state, and contains no thermal energy. The temperatures and were defined as those of the triple point of water. This definition served the following purposes: it fixed 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 established that one kelvin has precisely the same magnitude as one degree on the Celsius scale; and it established the difference between the null points of these scales as being ( = and = ). Since 2019, there has been a new definition based on the Boltzmann constant, Definition agreed by the 26th General Conference on Weights and Measures (CGPM) in November 2018, implemented 20 May 2019 but the scales are scarcely changed.
In the United States, the Fahrenheit scale is the most widely used. On this scale the freezing point of water corresponds to and the boiling point to . The Rankine scale, still used in fields of chemical engineering in the US, is an absolute scale based on the Fahrenheit increment.
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 quantummechanical oscillators and considers the system as a statistical ensemble of microstates. As a collection of classical material particles, the temperature is a measure of the mean energy of motion, called translational kinetic energy, of the particles, whether in solids, liquids, gases, or plasmas. The kinetic energy, a concept of classical mechanics, is half the mass of a particle times its speed squared. 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 monatomic and, approximately, in most gas and in simple metals, the temperature is a measure of the mean particle translational kinetic energy, 3/2 k_{B} T. It also determines the probability distribution function of 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.
Kinetic energy is also considered as a component of 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 depends on the temperature, i.e. the energy region of the interactions under consideration. For solids, the thermal energy is associated primarily with the Atom vibrations of its atoms or molecules about their equilibrium position. In an ideal gas, the kinetic energy is found exclusively in the purely translational motions of the particles. In other systems, and motions also contribute degrees of freedom.
The ideal gas law is based on observed empirical relationships between pressure ( p), volume ( V), and temperature ( T), and was recognized long before the kinetic theory of gases was developed (see Boyle's and Charles's laws). The ideal gas law states: The Feynman Lectures on Physics. 39–5 The ideal gas law
This relationship gives us our first hint that there is an absolute zero on the temperature scale, because it only holds if the temperature is measured on an absolute scale such as Kelvin's. The ideal gas law allows one to measure temperature on this absolute scale using the gas thermometer. The temperature in kelvins can be defined as the pressure in pascals of one mole of gas in a container of one cubic meter, divided by the gas constant.
Although it is not a particularly convenient device, the gas thermometer provides an essential theoretical basis by which all thermometers can be calibrated. As a practical matter, it is not possible to use a gas thermometer to measure absolute zero temperature since the gases 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, as shown in the figure.
The kinetic theory assumes that pressure is caused by the force associated with individual atoms striking the walls, and that all energy is translational kinetic energy. Using a sophisticated symmetry argument, Ludwig Boltzmann deduced what is now called the Maxwell–Boltzmann probability distribution function for the velocity of particles in an ideal gas. From that probability distribution function, the average kinetic energy (per particle) of a monatomic ideal gas isTolman, R.C. (1938). The Principles of Statistical Mechanics, Oxford University Press, London, pp. 93, 655.
where the Boltzmann constant is the ideal gas constant divided by the Avogadro number, and $v\_\backslash text\{rms\}\; =\; \backslash sqrt\{\backslash langle\; v^2\; \backslash rangle\}\; =\; \backslash sqrt\{\backslash langle\; \backslash mathbf\{v\backslash cdot\; v\}\; \backslash rangle\}$ is the rootmeansquare speed. This direct proportionality between temperature and mean molecular kinetic energy is a special case of the equipartition theorem, and holds only in the classical limit of a perfect gas. It does not hold exactly for most substances.
One 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.Bailyn, M. (1994). A Survey of Thermodynamics, American Institute of Physics Press, New York, , p. 22.Guggenheim, E.A. (1967). Thermodynamics. An Advanced Treatment for Chemists and Physicists, Elsevier, Amsterdam, (1st edition 1949) fifth edition 1965, p. 8: "If two systems are both in thermal equilibrium with a third system then they are in thermal equilibrium with each other."Buchdahl, H.A. (1966). The Concepts of Classical Thermodynamics, Cambridge University Press, Cambridge, p. 29: "... if each of two systems is in equilibrium with a third system then they are in equilibrium with each other."
This statement helps to define temperature but it does not, by itself, complete the definition. An empirical temperature is a numerical scale for the hotness of a thermodynamic system. Such hotness may be defined as existing on a onedimensional manifold, stretching between hot and cold. Sometimes the zeroth law is stated to include the existence of a unique universal hotness manifold, and of numerical scales on it, so as to provide a complete definition of empirical temperature. To be suitable for empirical thermometry, a material must have a monotonic relation between hotness and some easily measured state variable, such as pressure or volume, when all other relevant coordinates are fixed. An exceptionally suitable system is the ideal gas, which can provide a temperature scale that matches the absolute Kelvin scale. The Kelvin scale is defined on the basis of the second law of thermodynamics.
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 the outcome is always 100% the same result. In contrast, many mixed ( disordered) outcomes are possible, and their number increases with each toss. Eventually, the combinations of ~50% heads and ~50% tails dominate, and obtaining an outcome significantly different from 50/50 becomes increasingly unlikely. Thus the system naturally progresses to a state of maximum disorder or entropy.
As temperature governs the transfer of heat between two systems and the universe tends to progress toward a maximum of entropy, it is expected that there is some relationship between temperature and entropy. A heat engine is a device for converting thermal energy into mechanical energy, resulting in the performance of work. An analysis of the Carnot heat engine provides the necessary relationships. According to energy conservation and energy being a state function that does not change over a full cycle, the work from a heat engine over a full cycle is equal to the net heat, i.e. the sum of the heat put into the system at high temperature, q_{H} > 0, and the waste heat given off at the low temperature, q_{C} < 0..
The efficiency is the work divided by the heat input:
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:
= f\left(T_\text{H}, T_\text{C}\right).5}}
Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, a heat engine operating between T_{1} and T_{3} must have the same efficiency as one consisting of two cycles, one between T_{1} and T_{2}, and the second between T_{2} and T_{3}. This can only be the case if
which implies
Since the first function is independent of T_{2}, this temperature must cancel on the right side, meaning f( T_{1}, T_{3}) is of the form g( T_{1})/ g( T_{3}) (i.e. = = = , where g is a function of a single temperature. A temperature scale can now be chosen with the property that
= \frac{T_\text{C}}{T_\text{H}}.6}}
Substituting (6) back into (4) gives a relationship for the efficiency in terms of temperature:
= 1  \frac{T_\text{C}}{T_\text{H}}.7}}
For T_{C} = 0K the efficiency is 100% and that efficiency becomes greater than 100% below 0K. Since an efficiency greater than 100% violates the first law of thermodynamics, this implies that 0K is the minimum possible temperature. In fact, the lowest temperature ever obtained in a macroscopic system was 20nK, which was achieved in 1995 at NIST. Subtracting the right hand side of (5) from the middle portion and rearranging gives
where the negative sign indicates heat ejected from the system. This relationship suggests the existence of a state function, S, whose change characteristically vanishes for a complete cycle if it is defined by
where the subscript indicates a reversible process. This function corresponds to the entropy of the system, which was described previously. Rearranging (8) gives a formula for temperature in terms of fictive infinitesimal quasireversible elements of entropy and heat:
For a constantvolume system where entropy S( E) is a function of its energy E, d E = d q_{rev} and (9) gives
i.e. the reciprocal of the temperature is the rate of increase of entropy with respect to energy at constant volume.
When two systems with different temperatures are put into purely thermal connection, heat will flow from the higher temperature system to the lower temperature one; thermodynamically this is understood by the second law of thermodynamics: The total change in entropy following a transfer of energy $\backslash Delta\; E$ from system 1 to system 2 is:
and is thus positive if $T\_1\; >\; T\_2$
From the point of view of statistical mechanics, the total number of microstates in the combined system 1 + system 2 is $N\_1\; \backslash cdot\; N\_2$, the logarithm of which (times the Boltzmann constant) is the sum of their entropies; thus a flow of heat from high to low temperature, which brings an increase in total entropy, is more likely than any other scenario (normally it is much more likely), as there are more microstates in the resulting macrostate.
The international kinetic theory temperature of a body cannot take negative values. The thermodynamic temperature scale, however, is not so constrained.
For a body of matter, there can sometimes be conceptually defined, in terms of microscopic degrees of freedom, namely particle spins, a subsystem, with a temperature other than that of the whole body. When the body is in its own state of internal thermodynamic equilibrium, the temperatures of the whole body and of the subsystem must be the same. The two temperatures can differ when, by work through externally imposed force fields, energy can be transferred to and from the subsystem, separately from the rest of the body; then the whole body is not in its own state of internal thermodynamic equilibrium. There is an upper limit of energy such a spin subsystem can attain.
Considering the subsystem to be in a temporary state of virtual thermodynamic equilibrium, it is possible to obtain a negative temperature on the thermodynamic scale. Thermodynamic temperature is the inverse of the derivative of the subsystem's entropy with respect to its internal energy. As the subsystem's internal energy increases, the entropy increases for some range, but eventually attains a maximum value and then begins to decrease as the highest energy states begin to fill. At the point of maximum entropy, the temperature function shows the behavior of a singularity, because the slope of the entropy as a function of energy decreases to zero and then turns negative. As the subsystem's entropy reaches its maximum, its thermodynamic temperature goes to positive infinity, switching to negative infinity as the slope turns negative. Such negative temperatures are hotter than any positive temperature. Over time, when the subsystem is exposed to the rest of the body, which has a positive temperature, energy is transferred as heat from the negative temperature subsystem to the positive temperature system.
The kinetic theory temperature is not defined for such subsystems.

