Positive thinking about negative temperatures or, negative absolute temperatures: facts and myths Oliver Penrose Department of Mathematics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh Conference in memory of Bernard Jancovici, Paris, Nov. 5-6, 2015
Outline ◮ Some experiments
Outline ◮ Some experiments ◮ Some thermodynamics
Outline ◮ Some experiments ◮ Some thermodynamics ◮ Some statistical mechanics
Outline ◮ Some experiments ◮ Some thermodynamics ◮ Some statistical mechanics ◮ Some myths
Does ‘Negative absolute temperature’ make any sense? NO ◮ everybody knows nothing can be colder than absolute zero YES
Does ‘Negative absolute temperature’ make any sense? NO ◮ everybody knows nothing can be colder than absolute zero ◮ various authorities assert T > 0 (as an axiom) YES
Does ‘Negative absolute temperature’ make any sense? NO ◮ everybody knows nothing can be colder than absolute zero ◮ various authorities assert T > 0 (as an axiom) ◮ various formulations of Second Law uncomfortable with T < 0 YES
Does ‘Negative absolute temperature’ make any sense? NO ◮ everybody knows nothing can be colder than absolute zero ◮ various authorities assert T > 0 (as an axiom) ◮ various formulations of Second Law uncomfortable with T < 0 ◮ Ideal gas thermometer T = pV / Nk . Gas pressure can’t be negative, so neither can T YES
Does ‘Negative absolute temperature’ make any sense? NO ◮ everybody knows nothing can be colder than absolute zero ◮ various authorities assert T > 0 (as an axiom) ◮ various formulations of Second Law uncomfortable with T < 0 ◮ Ideal gas thermometer T = pV / Nk . Gas pressure can’t be negative, so neither can T ◮ The Hamiltonian is normally unbounded above, so that � e − E i / kT Z = i normally diverges if T ≤ 0 YES
Does ‘Negative absolute temperature’ make any sense? NO ◮ everybody knows nothing can be colder than absolute zero ◮ various authorities assert T > 0 (as an axiom) ◮ various formulations of Second Law uncomfortable with T < 0 ◮ Ideal gas thermometer T = pV / Nk . Gas pressure can’t be negative, so neither can T ◮ The Hamiltonian is normally unbounded above, so that � e − E i / kT Z = i normally diverges if T ≤ 0 YES ◮ BUT ... suppose energy is bounded above (e.g. Ising model)
Does ‘Negative absolute temperature’ make any sense? NO ◮ everybody knows nothing can be colder than absolute zero ◮ various authorities assert T > 0 (as an axiom) ◮ various formulations of Second Law uncomfortable with T < 0 ◮ Ideal gas thermometer T = pV / Nk . Gas pressure can’t be negative, so neither can T ◮ The Hamiltonian is normally unbounded above, so that � e − E i / kT Z = i normally diverges if T ≤ 0 YES ◮ BUT ... suppose energy is bounded above (e.g. Ising model) ◮ then the sum for Z makes sense even for negative T .
Does ‘Negative absolute temperature’ make any sense? NO ◮ everybody knows nothing can be colder than absolute zero ◮ various authorities assert T > 0 (as an axiom) ◮ various formulations of Second Law uncomfortable with T < 0 ◮ Ideal gas thermometer T = pV / Nk . Gas pressure can’t be negative, so neither can T ◮ The Hamiltonian is normally unbounded above, so that � e − E i / kT Z = i normally diverges if T ≤ 0 YES ◮ BUT ... suppose energy is bounded above (e.g. Ising model) ◮ then the sum for Z makes sense even for negative T . ◮ Example: for a nuclear spin µ in a magnetic field h Z = e µ h / kT + e − µ h / kT = 2 cosh( µ | h | / kT )
‘Canonical’ probabilities for a single spin- 1 2 nucleus ◮ Suppose magnetic moment of nucleus has magnitude µ .
‘Canonical’ probabilities for a single spin- 1 2 nucleus ◮ Suppose magnetic moment of nucleus has magnitude µ . ◮ Directed magnetic moment (plus sign means “parallel to h ”) is either + µ with energy − µ | h | and probability ∝ e µ | h | / kT − µ with energy + µ | h | and probability ∝ e − µ | h | / kT or
‘Canonical’ probabilities for a single spin- 1 2 nucleus ◮ Suppose magnetic moment of nucleus has magnitude µ . ◮ Directed magnetic moment (plus sign means “parallel to h ”) is either + µ with energy − µ | h | and probability ∝ e µ | h | / kT − µ with energy + µ | h | and probability ∝ e − µ | h | / kT or ◮ For positive T the lower-energy state is the more probable; for negative T the higher-energy state is the more probable: negative T is hotter than positive, not colder.
‘Canonical’ probabilities for a single spin- 1 2 nucleus ◮ Suppose magnetic moment of nucleus has magnitude µ . ◮ Directed magnetic moment (plus sign means “parallel to h ”) is either + µ with energy − µ | h | and probability ∝ e µ | h | / kT − µ with energy + µ | h | and probability ∝ e − µ | h | / kT or ◮ For positive T the lower-energy state is the more probable; for negative T the higher-energy state is the more probable: negative T is hotter than positive, not colder. ◮ Expectation magnetic moment of an N -spin system is N µ e µ | h | / kT − µ e − µ | h | / kT kT ≈ N µ 2 = N µ tanh µ | h | kT | h | e µ | h | / kT + e − µ | h | / kT along direction of h ; i.e., M is parallel to h at positive temperatures, but antiparallel at negative temps.
‘Canonical’ probabilities for a single spin- 1 2 nucleus ◮ Suppose magnetic moment of nucleus has magnitude µ . ◮ Directed magnetic moment (plus sign means “parallel to h ”) is either + µ with energy − µ | h | and probability ∝ e µ | h | / kT − µ with energy + µ | h | and probability ∝ e − µ | h | / kT or ◮ For positive T the lower-energy state is the more probable; for negative T the higher-energy state is the more probable: negative T is hotter than positive, not colder. ◮ Expectation magnetic moment of an N -spin system is N µ e µ | h | / kT − µ e − µ | h | / kT kT ≈ N µ 2 = N µ tanh µ | h | kT | h | e µ | h | / kT + e − µ | h | / kT along direction of h ; i.e., M is parallel to h at positive temperatures, but antiparallel at negative temps. ◮ Curie’s law M ≈ const kT h
Some experiments ◮ The Purcell-Pound experiment: first creation of a ‘negative-temperature’ state (1951)
Some experiments ◮ The Purcell-Pound experiment: first creation of a ‘negative-temperature’ state (1951) ◮ A nuclear spin system, normally antiferomagnetic, showing ferromagnetic ordering in the ‘negative temperature’ state (1992)
Some experiments ◮ The Purcell-Pound experiment: first creation of a ‘negative-temperature’ state (1951) ◮ A nuclear spin system, normally antiferomagnetic, showing ferromagnetic ordering in the ‘negative temperature’ state (1992) ◮ A lattice system showing Bose-Einstein condensation into the highest single-particle energy level (2013)
Purcell-Pound ∗ experiment on a paramagnetic crystal (LiF) Think of the crystal as a system of nuclear spins. At sufficiently low temperatures its relaxation time for spin-lattice interactions is of order 10 seconds, but for spin-spin interactions is of order 10 millisec ◮ 1 : bring to equilibrium with the lattice at a low temperature in a strong magnetic field. Time taken ≫ 10 sec ∗ E. M. Purcell & R. V. Pound, A nuclear spin system at negative temperatures Phys. Rev. 81, 279 (1951)
Purcell-Pound ∗ experiment on a paramagnetic crystal (LiF) Think of the crystal as a system of nuclear spins. At sufficiently low temperatures its relaxation time for spin-lattice interactions is of order 10 seconds, but for spin-spin interactions is of order 10 millisec ◮ 1 : bring to equilibrium with the lattice at a low temperature in a strong magnetic field. Time taken ≫ 10 sec ◮ 2 : Remove the strong field (this cools the spin system to an even lower temperature), apply a small oscillating field h of period ∼ 10 msec. or greater. M follows the oscillations and is parallel to h ∗ E. M. Purcell & R. V. Pound, A nuclear spin system at negative temperatures Phys. Rev. 81, 279 (1951)
Purcell-Pound ∗ experiment on a paramagnetic crystal (LiF) Think of the crystal as a system of nuclear spins. At sufficiently low temperatures its relaxation time for spin-lattice interactions is of order 10 seconds, but for spin-spin interactions is of order 10 millisec ◮ 1 : bring to equilibrium with the lattice at a low temperature in a strong magnetic field. Time taken ≫ 10 sec ◮ 2 : Remove the strong field (this cools the spin system to an even lower temperature), apply a small oscillating field h of period ∼ 10 msec. or greater. M follows the oscillations and is parallel to h ◮ 3 : reverse the magnetic field in a time ( ≪ 10 µ sec) so short that M cannot follow ∗ E. M. Purcell & R. V. Pound, A nuclear spin system at negative temperatures Phys. Rev. 81, 279 (1951)
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