negative temperature

temperature/coldness conversion scale: Temperatures on the Kelvin scale are shown in blue (Celsius scale in green, Fahrenheit scale in red), coldness values in gigabyte per nanojoule are shown in black. Infinite temperature (coldness zero) is shown at the top of the diagram; positive values of coldness/temperature are on the right-hand side, negative values on the left-hand side. Certain thermodynamic systems can achieve negative thermodynamic temperature; that is, their temperature can be expressed as a negative quantity on the Kelvin or Rankine scales. This should be distinguished from temperatures expressed as negative numbers on non-thermodynamic Celsius or Fahrenheit scales, which are nevertheless higher than absolute zero. Temperature is related to how particles are distributed among the available energy states of a system. At ordinary positive temperatures, lower-energy states are more likely to be occupied than higher-energy states. At negative temperatures, this distribution is reversed, so higher-energy states become more populated than lower-energy ones. For this reason, a system with a truly negative temperature on the Kelvin scale is hotter than any system with a positive temperature. If a negative-temperature system and a positive-temperature system come in contact, heat will flow from the negative- to the positive-temperature system.…

temperature/coldness conversion scale: Temperatures on the Kelvin scale are shown in blue (Celsius scale in green, Fahrenheit scale in red), coldness values in gigabyte per nanojoule are shown in black. Infinite temperature (coldness zero) is shown at the top of the diagram; positive values of coldness/temperature are on the right-hand side, negative values on the left-hand side. Certain thermodynamic systems can achieve negative thermodynamic temperature; that is, their temperature can be expressed as a negative quantity on the Kelvin or Rankine scales. This should be distinguished from temperatures expressed as negative numbers on non-thermodynamic Celsius or Fahrenheit scales, which are nevertheless higher than absolute zero. Temperature is related to how particles are distributed among the available energy states of a system. At ordinary positive temperatures, lower-energy states are more likely to be occupied than higher-energy states. At negative temperatures, this distribution is reversed, so higher-energy states become more populated than lower-energy ones. For this reason, a system with a truly negative temperature on the Kelvin scale is hotter than any system with a positive temperature. If a negative-temperature system and a positive-temperature system come in contact, heat will flow from the negative- to the positive-temperature system. A standard example of such a system is population inversion in laser physics. Temperature is also related to how a system's entropy changes as energy is added. Thermodynamic systems with no upper limit on their energy cannot achieve negative temperatures. This is the case for ordinary particles, such as atoms or dust, whose kinetic energy can increase without bound. In such systems, adding energy always increases the number of accessible states and therefore the entropy. Negative temperatures are only possible in systems that have a maximum energy they can hold. As such systems approach this maximum energy, the number of accessible states begins to decrease with further increases in energy, causing the entropy to fall and making the temperature negative. Examples of such systems include two-level systems and nuclear spin ensembles in an external magnetic field. T = \frac{dQ_\mathrm{rev}}{dS}. Entropy being a state function, the integral of over any cyclical process is zero. For a system in which the entropy is purely a function of the system's energy , the temperature can be defined as: T = \left( \frac{dS}{dE} \right)^{-1}. Equivalently, thermodynamic beta, or "coldness", is defined as \beta = \frac{1}{kT} = \frac{1}{k} \frac{dS}{dE}, where is the Boltzmann constant. Note that in classical thermodynamics, is defined in terms of temperature. This is reversed here, is the statistical entropy, a function of the possible microstates of the system, and temperature conveys information on the distribution of energy levels among the possible microstates. For systems with many degrees of freedom, the statistical and thermodynamic definitions of entropy are generally consistent with each other. Some theorists have proposed using an alternative definition of entropy as a way to resolve perceived inconsistencies between statistical and thermodynamic entropy for small systems and systems where the number of states decreases with energy, and the temperatures derived from these entropies are different. It has been argued that the new definition would create other inconsistencies; its proponents have argued that this is only apparent. T = \frac{dq_\mathrm{rev}}{dS} The relationship suggests that a positive temperature corresponds to the condition where entropy, , increases as thermal energy, , is added to the system. This is the "normal" condition in the macroscopic world, and is always the case for the translational, vibrational, rotational, and non-spin-related electronic and nuclear modes. The reason for this is that there are an infinite number of these types of modes, and adding more heat to the system increases the number of modes that are energetically accessible, and thus increases the entropy. E = \varepsilon\sum_{i=1}^N \sigma_i = \varepsilon j where is the sign of the th particle and is the number of particles with positive energy minus the number of particles with negative energy. From elementary combinatorics, the total number of microstates with this amount of energy is a binomial coefficient: \Omega_E = \binom{N}\frac{N + j}{2} = \frac{N!}{\left(\frac{N + j}{2}\right)! \left(\frac{N - j}{2}\right)!}. By the fundamental assumption of statistical mechanics, the entropy of this microcanonical ensemble is S = k_\text{B} \ln \Omega_E We can solve for thermodynamic beta ( ) by considering it as a central difference without taking the continuum limit: \begin{align} \beta &= \frac{1}{k_\mathrm{B}} \frac{\delta_{2\varepsilon}[S]}{2\varepsilon}\\[3pt] &= \frac{1}{2\varepsilon} \left( \ln \Omega_{E+\varepsilon} - \ln \Omega_{E-\varepsilon} \right)\\[3pt] &= \frac{1}{2\varepsilon} \ln \left( \frac{\left(\frac{N + j - 1}{2}\right)! \left(\frac{N - j + 1}{2}\right)!}{\left(\frac{N + j + 1}{2}\right)! \left(\frac{N - j - 1}{2}\right)!} \right)\\[3pt] &= \frac{1}{2\varepsilon} \ln \left( \frac{N - j + 1}{N + j + 1} \right). \end{align} hence the temperature T(E) = \frac{2\varepsilon}{k_\text{B}}\left[\ln \left( \frac{(N + 1)\varepsilon - E}{(N + 1)\varepsilon + E} \right)\right]^{-1}. This entire proof assumes the microcanonical ensemble with energy fixed and temperature being the emergent property. In the canonical ensemble, the temperature is fixed and energy is the emergent property. This leads to ( refers to microstates): \begin{align} Z(T) &= \sum_{i=1}^N e^{-\varepsilon_i\beta}\\[6pt] E(T) &= \frac{1}{Z}\sum_{i=1}^N \varepsilon_i e^{-\varepsilon_i\beta}\\[6pt] S(T) &= k_\text{B}\ln(Z) + \frac{E}{T} \end{align} Following the previous example, we choose a state with two levels and two particles. This leads to microstates , , , and . \begin{align} Z(T) &= e^{-0\beta} + 2e^{-1\beta} + e^{-2\beta}\\[3pt] &= 1 + 2e^{-\beta} + e^{-2\beta}\\[6pt] E(T) &= \frac{0e^{-0\beta} + 2 \times 1e^{-1\beta} + 2e^{-2\beta}}{Z}\\[3pt] &= \frac{2e^{-\beta} + 2e^{-2\beta}}{Z}\\[3pt] &= \frac{2e^{-\beta} + 2e^{-2\beta}}{1 + 2e^{-\beta} + e^{-2\beta}}\\[6pt] S(T) &= k_\text{B}\ln\left(1 + 2e^{-\beta} + e^{-2\beta}\right) + \frac{2e^{-\beta} + 2e^{-2\beta}}{\left(1 + 2e^{-\beta} + e^{-2\beta}\right)T} \end{align} The resulting values for , , and all increase with and never need to enter a negative temperature regime. H = (h\nu - \mu)a^\dagger a. The density operator in the grand canonical ensemble is \rho = \frac{e^{-\beta H}}{\operatorname{Tr}\left(e^{-\beta H}\right)}. For the system to have a ground state, the trace to converge, and the density operator to be generally meaningful, must be positive semidefinite. So if , and is negative semidefinite, then must itself be negative, implying a negative temperature.

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