2884:
additional energy reduces the entropy, since it moves the system further from a 50/50 mixture. This reduction in entropy with the addition of energy corresponds to a negative temperature. In NMR spectroscopy, this corresponds to pulses with a pulse width of over 180° (for a given spin). While relaxation is fast in solids, it can take several seconds in solutions and even longer in gases and in ultracold systems; several hours were reported for silver and rhodium at picokelvin temperatures. It is still important to understand that the temperature is negative only with respect to nuclear spins. Other degrees of freedom, such as molecular vibrational, electronic and electron spin levels are at a positive temperature, so the object still has positive sensible heat. Relaxation actually happens by exchange of energy between the nuclear spin states and other states (e.g. through the
1774:
2823:
1340:
2228:
1769:{\displaystyle {\begin{aligned}\beta &={\frac {1}{k_{\mathrm {B} }}}{\frac {\delta _{2\varepsilon }}{2\varepsilon }}\\&={\frac {1}{2\varepsilon }}\left(\ln \Omega _{E+\varepsilon }-\ln \Omega _{E-\varepsilon }\right)\\&={\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)\\&={\frac {1}{2\varepsilon }}\ln \left({\frac {N-j+1}{N+j+1}}\right).\end{aligned}}}
2818:{\displaystyle {\begin{aligned}Z(T)&=e^{-0\beta }+2e^{-1\beta }+e^{-2\beta }\\&=1+2e^{-\beta }+e^{-2\beta }\\E(T)&={\frac {0e^{-0\beta }+2\times 1e^{-1\beta }+2e^{-2\beta }}{Z}}\\&={\frac {2e^{-\beta }+2e^{-2\beta }}{Z}}\\&={\frac {2e^{-\beta }+2e^{-2\beta }}{1+2e^{-\beta }+e^{-2\beta }}}\\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{aligned}}}
875:
853:
864:
97:. The possibility of a decrease in entropy as energy increases requires the system to "saturate" in entropy. This is only possible if the number of high energy states is limited. For a system of ordinary (quantum or classical) particles such as atoms or dust, the number of high energy states is unlimited (particle momenta can in principle be increased indefinitely). Some systems, however (see the
899:
640:
498:
22:
2177:
738:. In this case, energy flows fairly rapidly among the spin states of interacting atoms, but energy transfer between the nuclear spins and other modes is relatively slow. Since the energy flow is predominantly within the spin system, it makes sense to think of a spin temperature that is distinct from the temperature associated to other modes.
155:(Kelvin) scale can be loosely interpreted as the average kinetic energy of the system's particles. The existence of negative temperature, let alone negative temperature representing "hotter" systems than positive temperature, would seem paradoxical in this interpretation. The paradox is resolved by considering the more rigorous definition of
578::+0 K (−273.15 °C), …, +100 K (−173.15 °C), …, +300 K (+26.85 °C), …, +1000 K (+726.85 °C), …, +∞ K (+∞ °C), −∞ K (−∞ °C), …, −1000 K (−1273.15 °C), …, −300 K (−573.15 °C), …, −100 K (−373.15 °C), …, −0 K (−273.15 °C).
465:, 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.
2871:
In the absence of a magnetic field, such a two-spin system would have maximum entropy when half the atoms are in the spin-up state and half are in the spin-down state, and so one would expect to find the system with close to an equal distribution of spins. Upon application of a magnetic field, some
558:
Negative temperatures can only exist in a system where there are a limited number of energy states (see below). As the temperature is increased on such a system, particles move into higher and higher energy states, and as the temperature increases, the number of particles in the lower energy states
125:
must also be bounded by the finite area. Bounded phase space is the essential property that allows for negative temperatures, and can occur in both classical and quantum systems. As shown by
Onsager, a system with bounded phase space necessarily has a peak in the entropy as energy is increased. For
3153:
The two-dimensional systems of vortices confined to a finite area can form thermal equilibrium states at negative temperature, and indeed negative temperature states were first predicted by
Onsager in his analysis of classical point vortices. Onsager's prediction was confirmed experimentally for a
625:
In many familiar physical systems, temperature is associated to the kinetic energy of atoms. Since there is no upper bound on the momentum of an atom, there is no upper bound to the number of energy states available when more energy is added, and therefore no way to get to a negative temperature.
468:
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
2883:
Since we started with over half the atoms in the spin-down state, this initially drives the system towards a 50/50 mixture, so the entropy is increasing, corresponding to a positive temperature. However, at some point, more than half of the spins are in the spin-up position. In this case, adding
1946:
3144:
regime, it is possible to go from a low entropy positive temperature state to a low entropy negative temperature state. In the negative temperature state, the atoms macroscopically occupy the maximum momentum state of the lattice. The negative temperature ensembles equilibrated and showed long
28:
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
133:
The limited range of states accessible to a system with negative temperature means that negative temperature is associated with emergent ordering of the system at high energies. For example in
Onsager's point-vortex analysis negative temperature is associated with the emergence of large-scale
2872:
of the atoms will tend to align so as to minimize the energy of the system, thus slightly more atoms should be in the lower-energy state (for the purposes of this example we will assume the spin-down state is the lower-energy state). It is possible to add energy to the spin system using
563:
for systems with limited states.) By injecting energy into these systems in the right fashion, it is possible to create a system in which there are more particles in the higher energy states than in the lower ones. The system can then be characterised as having a negative temperature.
1228:
2868:, meaning that they correspond to the same energy. When an external magnetic field is applied, the energy levels are split, since those spin states that are aligned with the magnetic field will have a different energy from those that are anti-parallel to it.
478:
137:
It seems negative temperatures were first found experimentally in 1951, when
Purcell and Pound observed evidence for them in the nuclear spins of a lithium fluoride crystal placed in a magnetic field, and then removed from this field. They wrote:
179:
of temperature, being the more fundamental quantity. Systems with a positive temperature will increase in entropy as one adds energy to the system, while systems with a negative temperature will decrease in entropy as one adds energy to the system.
830:, 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
73:
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
3073:
2172:{\displaystyle {\begin{aligned}Z(T)&=\sum _{i=1}^{N}e^{-\varepsilon _{i}\beta }\\E(T)&={\frac {1}{Z}}\sum _{i=1}^{N}\varepsilon _{i}e^{-\varepsilon _{i}\beta }\\S(T)&=k_{\text{B}}\ln(Z)+{\frac {E}{T}}\end{aligned}}}
2233:
1345:
1048:
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1927:
1951:
142:
A system in a negative temperature state is not cold, but very hot, giving up energy to any system at positive temperature put into contact with it. It decays to a normal state through infinite temperature.
3226:
1090:
4045:
Gauthier, G.; Reeves, M. T.; Yu, X.; Bradley, A. S.; Baker, M. A.; Bell, T. A.; Rubinsztein-Dunlop, H.; Davis, M. J.; Neely, T. W. (2019). "Giant vortex clusters in a two-dimensional quantum fluid".
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Johnstone, S. P.; Groszek, A. J.; Starkey, P. T.; Billinton, C. J.; Simula, T. P.; Helmerson, K. (2019). "Evolution of large-scale flow from turbulence in a two-dimensional superfluid".
134:
clusters of vortices. This spontaneous ordering in equilibrium statistical mechanics goes against common physical intuition that increased energy leads to increased disorder.
481:
57:
scales. This phenomenon was first discovered at the
University of Alberta. This should be distinguished from temperatures expressed as negative numbers on non-thermodynamic
723:
However, in some situations, it is possible to isolate one or more of the modes. In practice, the isolated modes still exchange energy with the other modes, but the
1234:
834:
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.
2852:
The previous example is approximately realized by a system of nuclear spins in an external magnetic field. This allows the experiment to be run as a variation of
3000:
3933:
3111:
720:
modes of a system determines the macroscopic temperature. In a "normal" system, thermal energy is constantly being exchanged between the various modes.
2853:
469:
different. It has been argued that the new definition would create other inconsistencies; its proponents have argued that this is only apparent.
101:
below), have a maximum amount of energy that they can hold, and as they approach that maximum energy their entropy actually begins to decrease.
4191:
3744:
3392:
3281:
920:
661:
519:
987:
368:
1785:
4438:
Shen, Jian-Qi (2003). "Anti-shielding Effect and
Negative Temperature in Instantaneously Reversed Electric Fields and Left-Handed Media".
2856:. In the case of electronic and nuclear spin systems, there are only a finite number of modes available, often just two, corresponding to
480:
1077:
4486:
3711:
1223:{\displaystyle \Omega _{E}={\binom {N}{\frac {N+j}{2}}}={\frac {N!}{\left({\frac {N+j}{2}}\right)!\left({\frac {N-j}{2}}\right)!}}.}
946:
687:
545:
114:
626:
However, in statistical mechanics, temperature can correspond to other degrees of freedom than just kinetic energy (see below).
2920:
1932:
This entire proof assumes the microcanonical ensemble with energy fixed and temperature being the emergent property. In the
924:
665:
523:
160:
294:
215:
25:
751:
3472:
Hanggi, Peter; Hilbert, Stefan; Dunkel, Jorn (2016). "Meaning of temperature in different thermostatistical ensembles".
3706:. World Scientific series in 20th century physics, v. 21. Singapore; River Edge, N.J.: World Scientific. p. 417.
909:
650:
508:
4383:
Schmidt, Harry; Mahler, Günter (2005). "Control of Local
Relaxation Behavior in Closed Bipartite Quantum Systems".
3078:
For the system to have a ground state, the trace to converge, and the density operator to be generally meaningful,
1247:
2933:
928:
913:
669:
654:
614:), runs continuously from low energy to high as +∞, …, 0, …, −∞. Because it avoids the abrupt jump from +∞ to −∞,
527:
512:
3187:
2885:
559:
and in the higher energy states approaches equality. (This is a consequence of the definition of temperature in
284:
over any cyclical process is zero. For a system in which the entropy is purely a function of the system's energy
184:
156:
42:
2182:
Following the previous example, we choose a state with two levels and two particles. This leads to microstates
4579:
2991:
3636:
Varga, Peter (1998). "Minimax games, spin glasses, and the polynomial-time hierarchy of complexity classes".
3924:
3416:
Dunkel, Jorn; Hilbert, Stefan (2013). "Consistent thermostatistics forbids negative absolute temperatures".
1238:
727:
of this exchange is much slower than for the exchanges within the isolated mode. One example is the case of
34:
622:. Although a system can have multiple negative temperature regions and thus have −∞ to +∞ discontinuities.
488:
When the temperature is negative, higher energy states are more likely to be occupied than low energy ones.
4611:
3663:
713:
701:
560:
4222:
Braun, S.; Ronzheimer, J. P.; Schreiber, M.; Hodgman, S. S.; Rom, T.; Bloch, I.; Schneider, U. (2013).
3822:
Braun, S.; Ronzheimer, J. P.; Schreiber, M.; Hodgman, S. S.; Rom, T.; Bloch, I.; Schneider, U. (2013).
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75:
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3167:
462:
4281:
Parihar, V.; Widom, A.; Srivastava, Y. (2006). "Thermal Time Scales in a Color Glass
Condensate".
3474:
Philosophical
Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
3126:
and changing the overall harmonic potential from trapping to anti-trapping, thus transforming the
3122:
atoms. This was done by tuning the interactions of the atoms from repulsive to attractive using a
3118:, upper bounds were placed on the kinetic energy, interaction energy and potential energy of cold
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1933:
1327:
611:
571:, but rather it is hotter than infinite temperature. As Kittel and Kroemer (p. 462) put it,
447:
359:
172:
118:
4567:
3533:
Frenkel, Daan; Warren, Patrick B. (2015-02-01). "Gibbs, Boltzmann, and negative temperatures".
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as energy increases, and high-energy states necessarily have negative
Boltzmann temperature.
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3249:
3207:
3190:(1956-07-01). "Thermodynamics and Statistical Mechanics at Negative Absolute Temperatures".
874:
62:
3068:{\displaystyle \rho ={\frac {e^{-\beta H}}{\operatorname {Tr} \left(e^{-\beta H}\right)}}.}
117:
confined within a finite area, and realized that since their positions are not independent
4175:
3928:
3115:
2873:
1331:
1069:
164:
46:
3704:
Spectroscopy with coherent radiation: selected papers of Norman F. Ramsey with commentary
883:
Entropy, thermodynamic beta, and temperature as a function of the energy for a system of
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3439:
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3245:
3203:
4171:
3887:
Montgomery, D. C. (1972). "Two-dimensional vortex motion and "negative temperatures"".
3777:
Hsu, W.; Barakat, R. (1992). "Statistics and thermodynamics of luminescent radiation".
3155:
3141:
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852:
735:
731:
728:
717:
277:
29:
coldness/temperature are on the right-hand side, negative values on the left-hand side.
4469:
3359:
Purcell, E. M.; Pound, R. V. (1951). "A nuclear spin system at negative temperature".
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in which the interaction term becomes negligible. The total energy of the system is
568:
198:
79:
66:
58:
54:
4430:
3931:(1974). "Negative Temperature States of Two-Dimensional Plasmas and Vortex Fluids".
3685:
3519:
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4375:
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1068:
is the number of particles with positive energy minus the number of particles with
863:
110:
4359:
959:
The simplest example, albeit a rather nonphysical one, is to consider a system of
3382:
3271:
4575:
1936:, the temperature is fixed and energy is the emergent property. This leads to (
978:
898:
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705:
639:
497:
152:
122:
86:
4414:
4312:
3140:. Performing this transformation adiabatically while keeping the atoms in the
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3572:
3402:
3337:
3291:
4535:
4257:
4137:
4076:
3857:
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3721:
3227:"Comment on: Negative Kelvin temperatures: some anomalies and a speculation"
977:
but are otherwise noninteracting. This can be understood as a limit of the
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21:
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191:
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3447:
1043:{\displaystyle E=\varepsilon \sum _{i=1}^{N}\sigma _{i}=\varepsilon j}
432:{\displaystyle \beta ={\frac {1}{kT}}={\frac {1}{k}}{\frac {dS}{dE}},}
3985:
3962:
1922:{\displaystyle T(E)={\frac {2\varepsilon }{k_{\text{B}}}}\left^{-1}.}
50:
4201:
Castle, J.; Emmerich, W.; Heikes, R.; Miller, R.; Rayne, J. (1965).
3253:
69:. A system with a truly negative temperature on the Kelvin scale is
4120:
4059:
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4240:
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3547:
3430:
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724:
476:
20:
4328:
Mosk, A. (2005). "Atomic Gases at Negative Kinetic Temperature".
4214:
3739:. West Sussex, England: John Wiley & Sons Ltd. p. 273.
2923:
for a single mode of a luminescent radiation field at frequency
2901:
126:
energies exceeding the value where the peak occurs, the entropy
109:
The possibility of negative temperatures was first predicted by
90:
4224:"Negative Absolute Temperature for Motional Degrees of Freedom"
3824:"Negative Absolute Temperature for Motional Degrees of Freedom"
3601:(1951-01-15). "A Nuclear Spin System at Negative Temperature".
892:
633:
582:
The corresponding inverse temperature scale, for the quantity
491:
567:
A substance with a negative temperature is not colder than
457:
is defined in terms of temperature. This is reversed here,
3110:
Negative temperatures have also been achieved in motional
3102:
must itself be negative, implying a negative temperature.
4514:
Carr, Lincoln D. (2013-01-04). "Negative Temperatures?".
2912:
lasers) are in excited states. This is referred to as a
2844:
and never need to enter a negative temperature regime.
963:
particles, each of which can take an energy of either
3467:
3465:
3384:
The Laws of Thermodynamics: A Very Short Introduction
3273:
The Laws of Thermodynamics: A Very Short Introduction
3003:
2936:
2231:
1949:
1788:
1343:
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1093:
990:
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371:
348:{\displaystyle T=\left({\frac {dS}{dE}}\right)^{-1}.}
297:
218:
266:{\displaystyle T={\frac {dQ_{\mathrm {rev} }}{dS}}.}
3737:
Spin Dynamics: Basics of Nuclear Magnetic Resonance
799:{\displaystyle T={\frac {dq_{\mathrm {rev} }}{dS}}}
3145:lifetimes in an anti-trapping harmonic potential.
3067:
2979:
2900:systems, wherein a large fraction of the system's
2817:
2171:
1921:
1768:
1285:
1222:
1042:
798:
431:
347:
265:
4203:Science by Degrees: Temperature from Zero to Zero
3308:Onsager, L. (1949). "Statistical Hydrodynamics".
1136:
1110:
3764:"Positive and negative picokelvin temperatures"
1235:fundamental assumption of statistical mechanics
573:
4488:Towards Quantum Magnetism with Ultracold Atoms
2896:This phenomenon can also be observed in many
1286:{\displaystyle S=k_{\text{B}}\ln \Omega _{E}}
700:The distribution of energy among the various
575:The temperature scale from cold to hot runs:
89:cannot achieve negative temperatures: adding
8:
4508:Negative temperature, at about 48min. 53sec.
3934:Proceedings of the Royal Society of London A
2980:{\displaystyle H=(h\nu -\mu )a^{\dagger }a.}
190:is a function of the change in the system's
3387:. Oxford University Press. pp. 10–14.
3276:. Oxford University Press. pp. 89–95.
927:. Unsourced material may be challenged and
668:. Unsourced material may be challenged and
526:. Unsourced material may be challenged and
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947:Learn how and when to remove this message
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688:Learn how and when to remove this message
546:Learn how and when to remove this message
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3158:in a Bose-Einstein condensate in 2019.
2854:nuclear magnetic resonance spectroscopy
453:Note that in classical thermodynamics,
473:Heat and molecular energy distribution
16:Physical systems hotter than any other
3084:must be positive semidefinite. So if
1296:We can solve for thermodynamic beta (
288:, the temperature can be defined as:
85:Thermodynamic systems with unbounded
65:, which are nevertheless higher than
7:
3225:Tremblay, André-Marie (1975-11-18).
925:adding citations to reliable sources
666:adding citations to reliable sources
524:adding citations to reliable sources
163:. This reveals the tradeoff between
887:noninteracting two-level particles.
813:corresponds to the condition where
113:in 1949. Onsager was investigating
4498:. ETH Zurich, ITS-MMS; Switzerland
4485:Ketterle, Wolfgang (22 Sep 2010).
1475:
1450:
1369:
1274:
1114:
1095:
843:Noninteracting two-level particles
779:
776:
773:
745:can be based on the relationship:
243:
240:
237:
121:from their momenta, the resulting
39:negative thermodynamic temperature
14:
4470:10.1238/Physica.Regular.068a00087
2904:(for chemical and gas lasers) or
2876:techniques. This causes atoms to
809:The relationship suggests that a
1080:with this amount of energy is a
897:
873:
862:
851:
638:
618:is considered more natural than
496:
362:, or "coldness", is defined as
3098:is negative semidefinite, then
821:, increases as thermal energy,
171:contained in the system, with "
2958:
2943:
2631:
2625:
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2030:
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1963:
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1:
4496:The Zurich Physics Colloquium
4360:10.1103/PhysRevLett.95.040403
3149:Two-dimensional vortex motion
98:
3990:Il Nuovo Cimento (1943-1954)
3911:10.1016/0375-9601(72)90302-7
2990:The density operator in the
4568:"−K: Negative Temperatures"
3986:"Statistical hydrodynamics"
3735:Levitt, Malcolm H. (2008).
3535:American Journal of Physics
3234:American Journal of Physics
3106:Motional degrees of freedom
2880:from spin-down to spin-up.
161:Boltzmann's entropy formula
4630:
4415:10.1103/PhysRevE.72.016117
4313:10.1103/PhysRevC.73.017901
3984:Onsager, L. (1949-03-01).
2886:nuclear Overhauser effect
2828:The resulting values for
1326:) by considering it as a
185:thermodynamic temperature
157:thermodynamic temperature
147:Definition of temperature
4580:University of Nottingham
3801:10.1103/PhysRevB.46.6760
3678:10.1103/PhysRevE.57.6487
3128:Bose-Hubbard Hamiltonian
2992:grand canonical ensemble
2864:, these spin states are
1940:refers to microstates):
630:Temperature and disorder
4536:10.1126/science.1232558
4330:Physical Review Letters
4258:10.1126/science.1227831
4138:10.1126/science.aat5793
4077:10.1126/science.aat5718
3858:10.1126/science.1227831
1239:microcanonical ensemble
93:always increases their
3955:10.1098/rspa.1974.0018
3623:10.1103/PhysRev.81.279
3504:10.1098/rsta.2015.0039
3212:10.1103/PhysRev.103.20
3069:
2981:
2860:. In the absence of a
2819:
2173:
2070:
1993:
1923:
1779:hence the temperature
1770:
1287:
1237:, the entropy of this
1224:
1076:, the total number of
1044:
1020:
800:
580:
489:
433:
349:
267:
45:can be expressed as a
30:
3070:
2982:
2858:spin up and spin down
2820:
2174:
2050:
1973:
1924:
1771:
1288:
1225:
1045:
1000:
801:
734:in a strong external
561:statistical mechanics
487:
434:
350:
268:
24:
3001:
2934:
2914:population inversion
2229:
1947:
1786:
1341:
1248:
1091:
1082:binomial coefficient
988:
921:improve this section
811:positive temperature
752:
662:improve this section
520:improve this section
369:
295:
216:
76:population inversion
4528:2013Sci...339...42C
4462:2003PhyS...68...87S
4407:2005PhRvE..72a6117S
4352:2005PhRvL..95d0403M
4305:2006PhRvC..73a7901P
4250:2013Sci...339...52B
4130:2019Sci...364.1267J
4114:(6447): 1267–1271.
4069:2019Sci...364.1264G
4053:(6447): 1264–1267.
4002:1949NCim....6S.279O
3947:1974RSPSA.336..257E
3903:1972PhLA...39....7M
3850:2013Sci...339...52B
3793:1992PhRvB..46.6760H
3660:1998PhRvE..57.6487V
3615:1951PhRv...81..279P
3557:2015AmJPh..83..163F
3496:2016RSPTA.37450039H
3440:2014NatPh..10...67D
3322:1949NCim....6S.279O
3246:1976AmJPh..44..994T
3204:1956PhRv..103...20R
3168:Negative resistance
2888:with other spins).
1330:without taking the
1072:. From elementary
1060:is the sign of the
463:statistical entropy
4566:Moriarty, Philip.
4207:Walker and Company
4010:10.1007/BF02780991
3480:(2064): 20150039.
3330:10.1007/BF02780991
3124:Feshbach resonance
3112:degrees of freedom
3065:
2977:
2840:all increase with
2815:
2813:
2169:
2167:
1934:canonical ensemble
1919:
1766:
1764:
1328:central difference
1283:
1220:
1040:
796:
612:Boltzmann constant
490:
448:Boltzmann constant
429:
360:thermodynamic beta
345:
280:, the integral of
263:
183:The definition of
119:degrees of freedom
31:
4385:Physical Review E
4289:(17901): 017901.
4283:Physical Review C
4193:978-0-7167-1088-2
3941:(1606): 257–271.
3890:Physics Letters A
3787:(11): 6760–6767.
3780:Physical Review B
3746:978-0-470-51117-6
3700:Ramsey, Norman F.
3638:Physical Review E
3565:10.1119/1.4895828
3448:10.1038/nphys2815
3394:978-0-19-957219-9
3283:978-0-19-957219-9
3060:
2809:
2648:
2616:
2519:
2462:
2163:
2134:
2048:
1896:
1824:
1821:
1753:
1705:
1676:
1666:
1631:
1594:
1559:
1518:
1436:
1411:
1375:
1264:
1215:
1205:
1176:
1134:
1133:
957:
956:
949:
794:
698:
697:
690:
556:
555:
548:
485:
424:
404:
391:
327:
258:
197:under reversible
63:Fahrenheit scales
41:; that is, their
4619:
4583:
4555:
4510:
4505:
4503:
4493:
4481:
4455:
4453:cond-mat/0302351
4434:
4400:
4398:quant-ph/0502181
4379:
4345:
4343:cond-mat/0501344
4324:
4298:
4277:
4243:
4218:
4197:
4182:(2nd ed.).
4158:
4157:
4123:
4103:
4097:
4096:
4062:
4042:
4036:
4035:
4033:
4032:
3981:
3975:
3974:
3921:
3915:
3914:
3884:
3878:
3877:
3843:
3819:
3813:
3812:
3774:
3768:
3767:
3760:
3751:
3750:
3732:
3726:
3725:
3696:
3690:
3689:
3671:
3653:
3651:cond-mat/9604030
3644:(6): 6487–6492.
3633:
3627:
3626:
3591:
3585:
3584:
3550:
3530:
3524:
3523:
3489:
3469:
3460:
3459:
3433:
3413:
3407:
3406:
3379:Atkins, Peter W.
3375:
3369:
3368:
3356:
3350:
3349:
3310:Il Nuovo Cimento
3305:
3296:
3295:
3268:Atkins, Peter W.
3264:
3258:
3257:
3231:
3222:
3216:
3215:
3184:
3156:quantum vortices
3139:
3101:
3097:
3093:
3083:
3074:
3072:
3071:
3066:
3061:
3059:
3058:
3054:
3053:
3027:
3026:
3011:
2986:
2984:
2983:
2978:
2970:
2969:
2926:
2843:
2839:
2835:
2831:
2824:
2822:
2821:
2816:
2814:
2810:
2808:
2804:
2800:
2799:
2798:
2780:
2779:
2752:
2751:
2750:
2729:
2728:
2712:
2707:
2703:
2702:
2701:
2683:
2682:
2650:
2649:
2646:
2617:
2615:
2614:
2613:
2595:
2594:
2572:
2571:
2570:
2549:
2548:
2532:
2524:
2520:
2515:
2514:
2513:
2492:
2491:
2475:
2467:
2463:
2458:
2457:
2456:
2435:
2434:
2407:
2406:
2387:
2362:
2361:
2343:
2342:
2315:
2311:
2310:
2292:
2291:
2270:
2269:
2221:
2211:
2201:
2191:
2178:
2176:
2175:
2170:
2168:
2164:
2156:
2136:
2135:
2132:
2103:
2102:
2098:
2097:
2080:
2079:
2069:
2064:
2049:
2041:
2016:
2015:
2011:
2010:
1992:
1987:
1939:
1928:
1926:
1925:
1920:
1915:
1914:
1906:
1902:
1901:
1897:
1895:
1869:
1843:
1825:
1823:
1822:
1819:
1813:
1805:
1775:
1773:
1772:
1767:
1765:
1758:
1754:
1752:
1735:
1718:
1706:
1704:
1693:
1685:
1681:
1677:
1675:
1671:
1667:
1662:
1645:
1636:
1632:
1627:
1610:
1603:
1599:
1595:
1590:
1573:
1564:
1560:
1555:
1538:
1531:
1519:
1517:
1506:
1498:
1494:
1490:
1489:
1488:
1464:
1463:
1437:
1435:
1424:
1416:
1412:
1410:
1402:
1392:
1391:
1378:
1376:
1374:
1373:
1372:
1359:
1325:
1324:
1322:
1321:
1310:
1307:
1292:
1290:
1289:
1284:
1282:
1281:
1266:
1265:
1262:
1229:
1227:
1226:
1221:
1216:
1214:
1210:
1206:
1201:
1190:
1181:
1177:
1172:
1161:
1154:
1146:
1141:
1140:
1139:
1129:
1118:
1113:
1103:
1102:
1067:
1064:th particle and
1063:
1059:
1049:
1047:
1046:
1041:
1030:
1029:
1019:
1014:
976:
969:
962:
952:
945:
941:
938:
932:
901:
893:
886:
877:
866:
855:
829:
820:
805:
803:
802:
797:
795:
793:
785:
784:
783:
782:
762:
741:A definition of
693:
686:
682:
679:
673:
642:
634:
621:
617:
609:
605:
604:
602:
601:
596:
593:
551:
544:
540:
537:
531:
500:
492:
486:
460:
456:
445:
438:
436:
435:
430:
425:
423:
415:
407:
405:
397:
392:
390:
379:
354:
352:
351:
346:
341:
340:
332:
328:
326:
318:
310:
287:
283:
276:Entropy being a
272:
270:
269:
264:
259:
257:
249:
248:
247:
246:
226:
208:
196:
189:
49:quantity on the
4629:
4628:
4622:
4621:
4620:
4618:
4617:
4616:
4587:
4586:
4565:
4562:
4522:(6115): 42–43.
4513:
4501:
4499:
4491:
4484:
4440:Physica Scripta
4437:
4382:
4327:
4280:
4221:
4200:
4194:
4180:Thermal Physics
4170:
4167:
4165:Further reading
4162:
4161:
4105:
4104:
4100:
4044:
4043:
4039:
4030:
4028:
3983:
3982:
3978:
3923:
3922:
3918:
3886:
3885:
3881:
3834:(6115): 52–55.
3821:
3820:
3816:
3776:
3775:
3771:
3762:
3761:
3754:
3747:
3734:
3733:
3729:
3714:
3698:
3697:
3693:
3635:
3634:
3630:
3603:Physical Review
3593:
3592:
3588:
3532:
3531:
3527:
3471:
3470:
3463:
3415:
3414:
3410:
3395:
3377:
3376:
3372:
3358:
3357:
3353:
3307:
3306:
3299:
3284:
3266:
3265:
3261:
3254:10.1119/1.10248
3240:(10): 994–995.
3229:
3224:
3223:
3219:
3192:Physical Review
3186:
3185:
3181:
3176:
3164:
3151:
3131:
3116:optical lattice
3108:
3099:
3095:
3085:
3079:
3039:
3035:
3028:
3012:
2999:
2998:
2961:
2932:
2931:
2924:
2894:
2874:radio frequency
2850:
2841:
2837:
2833:
2829:
2812:
2811:
2784:
2768:
2758:
2754:
2753:
2736:
2717:
2713:
2687:
2671:
2661:
2657:
2641:
2634:
2619:
2618:
2599:
2583:
2573:
2556:
2537:
2533:
2522:
2521:
2499:
2480:
2476:
2465:
2464:
2442:
2420:
2392:
2388:
2379:
2364:
2363:
2347:
2331:
2313:
2312:
2296:
2277:
2255:
2248:
2227:
2226:
2219:
2213:
2209:
2203:
2199:
2193:
2189:
2183:
2166:
2165:
2127:
2120:
2105:
2104:
2089:
2081:
2071:
2033:
2018:
2017:
2002:
1994:
1966:
1945:
1944:
1937:
1870:
1844:
1838:
1831:
1827:
1826:
1814:
1806:
1784:
1783:
1763:
1762:
1736:
1719:
1713:
1697:
1683:
1682:
1646:
1640:
1611:
1605:
1604:
1574:
1568:
1539:
1533:
1532:
1526:
1510:
1496:
1495:
1474:
1449:
1442:
1438:
1428:
1414:
1413:
1403:
1380:
1379:
1363:
1351:
1339:
1338:
1332:continuum limit
1317:
1311:
1308:
1305:
1304:
1302:
1297:
1273:
1257:
1246:
1245:
1191:
1185:
1162:
1156:
1155:
1147:
1119:
1108:
1094:
1089:
1088:
1070:negative energy
1065:
1061:
1058:
1054:
1021:
986:
985:
971:
964:
960:
953:
942:
936:
933:
918:
902:
891:
890:
889:
888:
884:
880:
879:
878:
869:
868:
867:
858:
857:
856:
845:
840:
828:
822:
818:
786:
767:
763:
750:
749:
694:
683:
677:
674:
659:
643:
632:
619:
615:
607:
597:
594:
591:
590:
588:
583:
579:
552:
541:
535:
532:
517:
501:
477:
475:
458:
454:
443:
416:
408:
383:
367:
366:
319:
311:
305:
304:
293:
292:
285:
281:
250:
231:
227:
214:
213:
207:
201:
194:
187:
165:internal energy
149:
107:
17:
12:
11:
5:
4627:
4626:
4623:
4615:
4614:
4609:
4604:
4599:
4589:
4588:
4585:
4584:
4561:
4560:External links
4558:
4557:
4556:
4511:
4482:
4435:
4380:
4325:
4296:hep-ph/0505199
4278:
4234:(6115): 52–5.
4219:
4198:
4192:
4166:
4163:
4160:
4159:
4098:
4037:
3996:(2): 279–287.
3976:
3925:Edwards, S. F.
3916:
3879:
3814:
3769:
3752:
3745:
3727:
3712:
3691:
3669:10.1.1.306.470
3628:
3609:(2): 279–280.
3595:Purcell, E. M.
3586:
3541:(2): 163–170.
3525:
3461:
3418:Nature Physics
3408:
3393:
3381:(2010-03-25).
3370:
3361:Physics Review
3351:
3316:(2): 279–287.
3297:
3282:
3270:(2010-03-25).
3259:
3217:
3188:Ramsey, Norman
3178:
3177:
3175:
3172:
3171:
3170:
3163:
3160:
3150:
3147:
3142:Mott insulator
3107:
3104:
3076:
3075:
3064:
3057:
3052:
3049:
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3015:
3009:
3006:
2988:
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2968:
2964:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2893:
2890:
2862:magnetic field
2849:
2846:
2826:
2825:
2807:
2803:
2797:
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2783:
2778:
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2110:
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2101:
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849:
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826:
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792:
789:
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766:
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736:magnetic field
696:
695:
646:
644:
637:
631:
628:
577:
554:
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419:
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403:
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358:Equivalently,
356:
355:
344:
339:
336:
331:
325:
322:
317:
314:
308:
303:
300:
278:state function
274:
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205:
148:
145:
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143:
106:
103:
15:
13:
10:
9:
6:
4:
3:
2:
4625:
4624:
4613:
4612:Laser science
4610:
4608:
4605:
4603:
4600:
4598:
4595:
4594:
4592:
4581:
4577:
4573:
4572:Sixty Symbols
4569:
4564:
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4559:
4553:
4549:
4545:
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4537:
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4391:(7): 016117.
4390:
4386:
4381:
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4361:
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4336:(4): 040403.
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4184:W. H. Freeman
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3929:Taylor, J. B.
3926:
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3908:
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3900:
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3713:9789810232504
3709:
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3412:
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3386:
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3380:
3374:
3371:
3367:(2): 279–280.
3366:
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3355:
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3335:
3331:
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2910:semiconductor
2907:
2903:
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2891:
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2859:
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2848:Nuclear spins
2847:
2845:
2805:
2801:
2795:
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2789:
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2781:
2776:
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2304:
2301:
2297:
2293:
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2274:
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2263:
2260:
2256:
2252:
2250:
2242:
2236:
2225:
2224:
2223:
2216:
2206:
2196:
2186:
2160:
2157:
2152:
2146:
2140:
2137:
2128:
2124:
2122:
2114:
2108:
2099:
2094:
2090:
2086:
2082:
2076:
2072:
2066:
2061:
2058:
2055:
2051:
2045:
2042:
2037:
2035:
2027:
2021:
2012:
2007:
2003:
1999:
1995:
1989:
1984:
1981:
1978:
1974:
1970:
1968:
1960:
1954:
1943:
1942:
1941:
1935:
1916:
1911:
1908:
1903:
1898:
1892:
1889:
1886:
1880:
1877:
1874:
1866:
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1860:
1854:
1851:
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1839:
1835:
1832:
1828:
1815:
1810:
1807:
1801:
1795:
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1781:
1780:
1759:
1755:
1749:
1746:
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1737:
1732:
1729:
1726:
1723:
1720:
1714:
1710:
1707:
1701:
1698:
1694:
1689:
1687:
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1672:
1668:
1663:
1659:
1656:
1653:
1650:
1647:
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1633:
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1624:
1621:
1618:
1615:
1612:
1606:
1600:
1596:
1591:
1587:
1584:
1581:
1578:
1575:
1569:
1565:
1561:
1556:
1552:
1549:
1546:
1543:
1540:
1534:
1527:
1523:
1520:
1514:
1511:
1507:
1502:
1500:
1491:
1485:
1482:
1479:
1471:
1468:
1465:
1460:
1457:
1454:
1446:
1443:
1439:
1432:
1429:
1425:
1420:
1418:
1407:
1404:
1396:
1388:
1385:
1381:
1364:
1360:
1355:
1353:
1348:
1337:
1336:
1335:
1333:
1329:
1320:
1314:
1300:
1278:
1270:
1267:
1258:
1254:
1251:
1244:
1243:
1242:
1240:
1236:
1217:
1211:
1207:
1202:
1198:
1195:
1192:
1186:
1182:
1178:
1173:
1169:
1166:
1163:
1157:
1151:
1148:
1142:
1130:
1126:
1123:
1120:
1115:
1104:
1099:
1087:
1086:
1085:
1083:
1079:
1075:
1074:combinatorics
1071:
1037:
1034:
1031:
1026:
1022:
1016:
1011:
1008:
1005:
1001:
997:
994:
991:
984:
983:
982:
980:
975:
968:
951:
948:
940:
930:
926:
922:
916:
915:
911:
906:This section
904:
900:
895:
894:
876:
865:
854:
842:
837:
835:
833:
825:
816:
812:
790:
787:
768:
764:
758:
755:
748:
747:
746:
744:
739:
737:
733:
730:
726:
721:
719:
715:
711:
707:
703:
702:translational
692:
689:
681:
671:
667:
663:
657:
656:
652:
647:This section
645:
641:
636:
635:
629:
627:
623:
613:
600:
586:
576:
572:
570:
569:absolute zero
565:
562:
550:
547:
539:
529:
525:
521:
515:
514:
510:
505:This section
503:
499:
494:
493:
472:
470:
466:
464:
451:
449:
426:
420:
417:
412:
409:
401:
398:
393:
387:
384:
380:
375:
372:
365:
364:
363:
361:
342:
337:
334:
329:
323:
320:
315:
312:
306:
301:
298:
291:
290:
289:
279:
260:
254:
251:
232:
228:
222:
219:
212:
211:
210:
204:
200:
199:heat transfer
193:
186:
181:
178:
174:
170:
166:
162:
158:
154:
151:The absolute
146:
141:
140:
139:
135:
131:
129:
124:
120:
116:
112:
104:
102:
100:
96:
92:
88:
83:
81:
80:laser physics
77:
72:
68:
67:absolute zero
64:
60:
56:
52:
48:
44:
40:
36:
27:
23:
19:
4571:
4519:
4515:
4507:
4500:. Retrieved
4487:
4446:(1): 87–97.
4443:
4439:
4388:
4384:
4333:
4329:
4286:
4282:
4231:
4227:
4202:
4179:
4111:
4107:
4101:
4050:
4046:
4040:
4029:. Retrieved
3993:
3989:
3979:
3938:
3932:
3919:
3894:
3888:
3882:
3831:
3827:
3817:
3784:
3778:
3772:
3736:
3730:
3703:
3694:
3641:
3637:
3631:
3606:
3602:
3599:Pound, R. V.
3589:
3538:
3534:
3528:
3477:
3473:
3421:
3417:
3411:
3383:
3373:
3364:
3360:
3354:
3313:
3309:
3272:
3262:
3237:
3233:
3220:
3198:(1): 20–28.
3195:
3191:
3182:
3152:
3136:
3132:
3120:potassium-39
3109:
3090:
3086:
3080:
3077:
2989:
2918:
2895:
2882:
2877:
2870:
2865:
2851:
2827:
2214:
2204:
2194:
2184:
2181:
1931:
1778:
1318:
1312:
1298:
1295:
1232:
1052:
973:
966:
958:
943:
934:
919:Please help
907:
823:
810:
808:
740:
722:
699:
684:
675:
660:Please help
648:
624:
598:
584:
581:
574:
566:
557:
542:
533:
518:Please help
506:
467:
452:
441:
357:
275:
202:
182:
176:
159:in terms of
150:
136:
132:
127:
111:Lars Onsager
108:
84:
70:
38:
37:can achieve
32:
18:
4597:Temperature
4576:Brady Haran
4176:Kroemer, H.
3114:. Using an
2921:Hamiltonian
1078:microstates
979:Ising model
743:temperature
706:vibrational
153:temperature
123:phase space
115:2D vortices
87:phase space
43:temperature
4591:Categories
4172:Kittel, C.
4121:1801.06952
4060:1801.06951
4031:2019-11-17
3897:(1): 7–8.
3487:1507.05713
3174:References
3154:system of
2866:degenerate
725:time scale
714:electronic
710:rotational
177:reciprocal
4607:Magnetism
4552:124095369
4478:118894011
4321:119090586
4241:1211.0545
4093:195750381
4026:186224016
4018:1827-6121
3971:120771020
3841:1211.0545
3664:CiteSeerX
3581:119179342
3573:0002-9505
3548:1403.4299
3431:1304.2066
3424:(1): 67.
3403:467748903
3346:186224016
3338:1827-6121
3292:467748903
3048:β
3045:−
3033:
3021:β
3018:−
3005:ρ
2967:†
2956:μ
2953:−
2950:ν
2906:electrons
2796:β
2790:−
2777:β
2774:−
2748:β
2742:−
2726:β
2723:−
2699:β
2693:−
2680:β
2677:−
2655:
2611:β
2605:−
2592:β
2589:−
2568:β
2562:−
2546:β
2543:−
2511:β
2505:−
2489:β
2486:−
2454:β
2448:−
2432:β
2426:−
2415:×
2404:β
2398:−
2359:β
2353:−
2340:β
2337:−
2308:β
2302:−
2289:β
2283:−
2267:β
2261:−
2141:
2100:β
2091:ε
2087:−
2073:ε
2052:∑
2013:β
2004:ε
2000:−
1975:∑
1909:−
1887:ε
1864:−
1861:ε
1836:
1811:ε
1724:−
1711:
1702:ε
1657:−
1651:−
1579:−
1550:−
1524:
1515:ε
1486:ε
1483:−
1476:Ω
1472:
1466:−
1461:ε
1451:Ω
1447:
1433:ε
1408:ε
1389:ε
1382:δ
1349:β
1275:Ω
1271:
1196:−
1096:Ω
1035:ε
1023:σ
1002:∑
998:ε
937:July 2024
908:does not
678:July 2024
649:does not
536:July 2024
507:does not
373:β
335:−
128:decreases
4578:for the
4544:23288530
4431:17987338
4423:16090046
4368:16090784
4266:23288533
4215:64023985
4178:(1980).
4146:31249055
4085:31249054
3866:23288533
3809:10002377
3722:38753008
3702:(1998).
3686:10964509
3520:39161351
3512:26903095
3456:16757018
3162:See also
838:Examples
832:infinite
173:coldness
99:examples
47:negative
33:Certain
4602:Entropy
4524:Bibcode
4516:Science
4492:(movie)
4458:Bibcode
4403:Bibcode
4376:1156732
4348:Bibcode
4301:Bibcode
4274:8207974
4246:Bibcode
4228:Science
4154:4948239
4126:Bibcode
4108:Science
4065:Bibcode
4047:Science
3998:Bibcode
3943:Bibcode
3899:Bibcode
3874:8207974
3846:Bibcode
3828:Science
3789:Bibcode
3656:Bibcode
3611:Bibcode
3553:Bibcode
3492:Bibcode
3436:Bibcode
3318:Bibcode
3242:Bibcode
3200:Bibcode
1323:
1303:
1233:By the
929:removed
914:sources
815:entropy
729:nuclear
718:nuclear
670:removed
655:sources
610:is the
606:(where
603:
589:
528:removed
513:sources
461:is the
446:is the
192:entropy
175:", the
169:entropy
105:History
95:entropy
59:Celsius
55:Rankine
35:systems
4550:
4542:
4476:
4429:
4421:
4374:
4366:
4319:
4272:
4264:
4213:
4190:
4152:
4144:
4091:
4083:
4024:
4016:
3969:
3961:
3872:
3864:
3807:
3743:
3720:
3710:
3684:
3666:
3579:
3571:
3518:
3510:
3454:
3401:
3391:
3344:
3336:
3290:
3280:
3094:, and
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