Abstract
We describe recent progress towards deriving the Fundamental Laws of thermodynamics (the 0th, 1st, and 2nd Law) from nonequilibrium quantum statistical mechanics in simple, yet physically relevant models. Along the way, we clarify some basic thermodynamic notions and discuss various reversible and irreversible thermodynamic processes from the point of view of quantum statistical mechanics.
Similar content being viewed by others
References
W. Abou-Salem, Nonequilibrium quantum statistical mechanics and thermodynamics, ETH-Diss. 16187 (2005).
W. Abou-Salem and J. Fröhlich, Adiabatic theorems and reversible isothermal processes, Lett. Math. Phys. 72:153–163 (2005).
W. Abou-Salem and J. Fröhlich, Cyclic thermodynamic processes and entropy production, to appear in J. Stat. Phys.
W. Abou-Salem and J. Fröhlich, Adiabatic theorems for quantum resonances, to appear in Commun. Math. Phys.
W. Abou-Salem and J. Frölhlich, in preparation.
H. Araki and E. Woods, Representations of the canonical commutation relations describing a non-relativistic infinite free bose gas. J. Math. Phys. 4:637–662 (1963).
H. Araki and W. Wyss, Representations of canonical anticommutation relations, Helv. Phys. Acta 37:136 (1964).
J. E. Avron and A. Elgart, Adiabatic theorem without a gap condition, Commun. Math. Phys. 203:445–463 (1999).
V. Bach, J. Fröhlich and I. M. Sigal, Return to Equilibrium. J. Math. Phys. 41(6):3985–4061 (2000).
F. Benetto, J. L. Lebowitz and L. Rey-Bellet, Fourier’s Law: A Challenge to Theorists, Math. Phys. 2000:128–150, Imperial College Press, London (2000).
D. D. Botvich, and V. A. Malyshev, Unitary equivalence of temperature dy for ideal and locally perturbed Fermi gas, Commun. Math. Phys. 61:209 (1978).
J. B. Boyling, An axiomatic approach to classical thermodynamics, Proc. R. Soc. London A 329:35–70 (1972).
O. Bratteli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics 1,2, Texts and Monographs in Physics (Springer-Verlag Berlin, 1987).
J. Dereziński and V. Jaksić, Spectral theory of Pauli-Fierz operators, Ann. Henri Poincare 4:739 (2003).
J. Dereziński and V. Jaksić, C.-A. Pillet, Perturbation theory of KMS states, Rev. Math. Phys. 15:447 (2003).
J. Fröhlich and M. Merkli, Another return of “return to equilibrium,” Commun. Math. Phys. 251:235–262 (2004).
J. Fröhlich and M. Merkli, Thermal lonization. Math. Phys. Anal. Geometry 7:239–287 (2004).
J. Fröhlich, M. Merkli and I. M. Sigal, Ionization of atoms in a thermal field, J. Stat. Phys. 116:311–359 (2004).
J. Fröhlich, M. Merkli and D. Ueltschi, Dissipative transport: thermal contacts and tunnelling junctions, Ann. Henri Poincare 4:897–945 (2003).
J. Fröhlich, M. Merkli, S. Schwarz and D. Ueltschi, Statistical Mechanics of Thermodynamic Processes, in A Garden of Quanta (World Sci. Publishing, River Edge, New Jersey, 2003), pp. 345–363.
R. Haag, Local Quantum Physics. Fields, Particles, Algebras. Text and Monographs in Physics (Springer-Verlag Berlin, 1992).
R. Haag, N. M. Hugenholtz and M. Winnink, On the equilibrium states in quantum statistical mechanics., Commun. Math. Phys. 5:215–236 (1967).
R. Haag, D. Kastler and E. Trych-Pohlmeyer, Stability and equilibrium states, Commun. Math. Phys. 38:213 (1974).
K. Hepp, Rigorous results on the s–d model of the Kondo effect. Solid State Commun. 8:2087–2090 (1970).
V. Jaksić and C. A. Pillet, On a model for quantum friction II. Fermi’s golden rule and dynamics at positive temperature, Commun. Math. Phys. 176:619–644 (1996).
V. Jaksić and C. A. Pillet, On a model for quantum friction III. ergodic properties of the Spin-Boson system, Commun. Math. Phys. 178:627–651 (1996).
V. Jaksić and C.-A. Pillet, Non- equilibrium steady states of finite quantum systems coupled to thermal reservoirs, Commun. Math. Phys. 226:131–162 (2002).
V. Jaksic, Y. Ogata and C.-A. Pillet, The Green-Kubo formula and the Onsager reciprocity relations in quantum statistical mechanics, [Texas mp-arc 2005 preprint].
V. Jaksic, Y. Ogata and C.-A. Pillet, Linear response theory in quantum statistical mechanics, [Texas mp-arc 2005 preprint].
E. Lieb and M.-B. Ruskai, Proof of the strong subadditivity of quantum-mechanical entropy, J. Math. Phys. 14: 1938–1941 (1973).
E. Lieb and J. Yngvason, The mathematical structure of the second law of thermodynamics, in Current Developments in Mathematics, 2001 (International Press, Cambridge, 2002).
M. Merkli, Positive commutator method in non-equilibrium statistical mechanics, Commun. Math. Phys. 223:327–362 (2001).
M. Merkli, Stability of equilibria with a condensate, Commun. Math. Phys. 257:621–640 (2005).
M. Merkli, M. Mück and I. M. Sigal, Instability of equilibrium states for coupled heat reservoirs at different temperatures [axiv:math-ph/0508005].
M. Merkli, M. Mück and I. M. Sigal, Theory of nonequilibrium stationary states as a theory of resonances. Existence and properties of NESS, [arxiv:math-ph/0603006].
D. Robinson, Return to equilibrium, Commun. Math. Phys. 31:171–189 (1973).
D. Ruelle, Statistical Mechanics. Rigorous Results, Reprint of the 1989 Edition (World Scientific Publishing Co., Inc., River Edge, NJ; Imperial College Press, London, 1999).
D. Ruelle, Entropy production in quantum spin systems, Comm. Math. Phys. 224(1):3–16 (2001).
D. Ruelle Natural nonequilibrium states in quantum statistical mechanics, J. Stat. Phys. 98(1–2):57–75 (2000).
S. Sakai, C ∗-Algebras and W ∗-Algebras (Berlin, Springer, 1971).
S. Teufel, A note on the adiabatic theorem, Lett. Math. Phys. 58:261–266 (2001).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Salem, W.K.A., Fröhlich, J. Status of the Fundamental Laws of Thermodynamics. J Stat Phys 126, 1045–1068 (2007). https://doi.org/10.1007/s10955-006-9222-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-006-9222-8