doi: 10.3934/dcdsb.2018214

Thermodynamical potentials of classical and quantum systems

1. 

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

2. 

Department of Mathematics, Indiana University, Bloomington, IN 47405, USA

3. 

School of Medical Informatics and Engineering, Southwest Medical University, Luzhou, Sichuan 646000, China

* Corresponding author: Shouhong Wang

Received  November 2017 Published  June 2018

Fund Project: The work was supported in part by the US National Science Foundation (NSF), the Office of Naval Research (ONR) and by the Chinese National Science Foundation (11771306)

The aim of the paper is to systematically introduce thermodynamic potentials for thermodynamic systems and Hamiltonian energies for quantum systems of condensates. The study is based on the rich previous work done by pioneers in the related fields. The main ingredients of the study consist of 1) SO(3) symmetry of thermodynamical potentials, 2) theory of fundamental interaction of particles, 3) the statistical theory of heat developed recently [23], 4) quantum rules for condensates that we postulate in Quantum Rule 4.1, and 5) the dynamical transition theory developed by Ma and Wang [20]. The statistical and quantum systems we study in this paper include conventional thermodynamic systems, thermodynamic systems of condensates, as well as quantum condensate systems. The potentials and Hamiltonian energies that we derive are based on first principles, and no mean-field theoretic expansions are used.

Citation: Ruikuan Liu, Tian Ma, Shouhong Wang, Jiayan Yang. Thermodynamical potentials of classical and quantum systems. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018214
References:
[1]

M. H. AndersonJ. R. EnsherM. R. MatthewsC. E. Wieman and E. A. Cornell, Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor, Collected Papers of Carl Wieman, (2008), 453-456. doi: 10.1142/9789812813787_0062.

[2]

S. N. Bose, Plancks Gesetz und Lichtquantenhypothese, Zeitschrift Für Physik, 26 (1924), 178-181. doi: 10.1007/BF01327326.

[3] P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics, Cambridge University Press, New York, 2000. doi: 10.1017/CBO9780511813467.
[4]

K. B. DavisM. O. MewesM. R. AndrewsN. J. van DrutenD. S. DurfeeD. M. Kurn and W. Ketterle, Bose-einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 75 (1996), 3969-3973. doi: 10.1109/EQEC.1996.561567.

[5]

P. de Gennes, Superconductivity of Metals and Alloys, W. A. Benjamin, 1966.

[6]

A. Einstein, Quantentheorie des einatomigen idealen gases, Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physik-Mathematik, (1924), 261-267.

[7]

A. Einstein, Quantentheorie des einatomigen idealen gases. zweite abhandlung., Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physik-Mathematik, (1925), 3-14.

[8]

M. E. Fisher, Renormalization group theory: Its basis and formulation in statistical physics, Reviews of Modern Physics, 70 (1998), 653-681. doi: 10.1103/RevModPhys.70.653.

[9]

V. L. Ginzburg, On superconductivity and super uidity (what i have and have not managed to do), as well as on the 'physical minimum' at the beginning of the xxi century, Phys.-Usp., 47 (2004), 1155-1170.

[10]

L. Gor'kov, Generalization of the Ginzburg-Landau equations for non-stationary problems in the case of alooys with paramagnetic impurities, Sov.Phys. JETP, 27 (1968), 328-334.

[11]

E. P. Gross, Structure of a quantized vortex in boson systems (1955-1965), Il Nuovo Cimento, 20 (1961), 454-477. doi: 10.1007/BF02731494.

[12]

T.-L. Ho, Spinor bose condensates in optical traps, Physical Review Letters, 81 (1998), 742. doi: 10.1103/PhysRevLett.81.742.

[13]

L. P. Kadanoff, Statistical Physics: Statics, Dynamics and Renormalization, World Scientifc Publishing Co Inc, 2000. doi: 10.1142/4016.

[14]

P. Kapitza, E7-Viscosity of liquid helium below the -point, Nature, 141 (1938), 74. doi: 10.1016/B978-0-08-015816-7.50011-0.

[15]

M. Kleman and O. D. Laverntovich, Soft Matter Physics: An Introduction, Springer Science Business Media, 2007. doi: 10.1007/b97416.

[16]

J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions in twodimensional systems, Journal of Physics C: Solid State Physics, 6 (1973), 1181. doi: 10.1088/0022-3719/6/7/010.

[17] L. D. Landau and E. M. Lifshitz, Statistical Physics: V. 5: Course of Theoretical Physics, Pergamon Press, 1969.
[18]

E. M. Lifschitz and L. P. Pitajewski, Lehrbuch Der Theoretischen Physik ("LandauLifschitz"). Band X, 2nd edition, Akademie-Verlag, Berlin, 1990. Physikalische Kinetik. [Physical kinetics], Translated from the Russian by Gerd Röpke and Thomas Frauenheim, Translation edited and with a foreword by Paul Ziesche and Gerhard Diener.

[19]

E. Lifshitz and L. Pitaevskii, Statistical Physics Part 2, Landau and Lifshitz Course of Theoretical Physics vol. 9, 1980.

[20] T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag,, New York, 2014. doi: 10.1007/978-1-4614-8963-4.
[21] T. Ma and S. Wang, Mathematical Principles of Theoretical Physics, Science Press, Beijing, 2015.
[22]

T. Ma and S. Wang, Dynamic law of physical motion and potential-descending principle, J. Math. Study, 50 (2017), 215-241. doi: 10.4208/jms.v50n3.17.02.

[23]

T. Ma and S. Wang, Statistical Theory of Heat, Hal preprint: Hal-01578634, 2017.

[24]

T. Ma and S. Wang, Topological phase transitions Ⅰ: Quantum phase transitions, to appear, (2018).

[25]

T. Ohmi and K. Machida, Bose-Einstein condensation with internal degrees of freedom in alkali atom gases, Journal of the Physical Society of Japan, 67 (1998), 1822-1825. doi: 10.1143/JPSJ.67.1822.

[26]

R. K. Pathria and P. D. Beale, Statistical Mechanics, 3nd edition, Elsevier, 2011.

[27]

L. Pitaevskii, Vortex lines in an imperfect bose gas, Sov. Phys. JETP, 13 (1961), 451-454.

[28]

L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation and Super Uidity, vol. 164 of Internat. Ser. Mono. Phys., Clarendon Press, Oxford, 2003.

[29]

L. E. Reichl, A modern Course in Statistical Physics, A Wiley-Interscience Publication, 2nd edition, John Wiley Sons Inc., New York, 1998.

[30]

H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, New York and Oxford, 1971.

[31]

M. Tinkham, Introduction to Superconductivity, McGraw-Hill, Inc, 1996.

show all references

References:
[1]

M. H. AndersonJ. R. EnsherM. R. MatthewsC. E. Wieman and E. A. Cornell, Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor, Collected Papers of Carl Wieman, (2008), 453-456. doi: 10.1142/9789812813787_0062.

[2]

S. N. Bose, Plancks Gesetz und Lichtquantenhypothese, Zeitschrift Für Physik, 26 (1924), 178-181. doi: 10.1007/BF01327326.

[3] P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics, Cambridge University Press, New York, 2000. doi: 10.1017/CBO9780511813467.
[4]

K. B. DavisM. O. MewesM. R. AndrewsN. J. van DrutenD. S. DurfeeD. M. Kurn and W. Ketterle, Bose-einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 75 (1996), 3969-3973. doi: 10.1109/EQEC.1996.561567.

[5]

P. de Gennes, Superconductivity of Metals and Alloys, W. A. Benjamin, 1966.

[6]

A. Einstein, Quantentheorie des einatomigen idealen gases, Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physik-Mathematik, (1924), 261-267.

[7]

A. Einstein, Quantentheorie des einatomigen idealen gases. zweite abhandlung., Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physik-Mathematik, (1925), 3-14.

[8]

M. E. Fisher, Renormalization group theory: Its basis and formulation in statistical physics, Reviews of Modern Physics, 70 (1998), 653-681. doi: 10.1103/RevModPhys.70.653.

[9]

V. L. Ginzburg, On superconductivity and super uidity (what i have and have not managed to do), as well as on the 'physical minimum' at the beginning of the xxi century, Phys.-Usp., 47 (2004), 1155-1170.

[10]

L. Gor'kov, Generalization of the Ginzburg-Landau equations for non-stationary problems in the case of alooys with paramagnetic impurities, Sov.Phys. JETP, 27 (1968), 328-334.

[11]

E. P. Gross, Structure of a quantized vortex in boson systems (1955-1965), Il Nuovo Cimento, 20 (1961), 454-477. doi: 10.1007/BF02731494.

[12]

T.-L. Ho, Spinor bose condensates in optical traps, Physical Review Letters, 81 (1998), 742. doi: 10.1103/PhysRevLett.81.742.

[13]

L. P. Kadanoff, Statistical Physics: Statics, Dynamics and Renormalization, World Scientifc Publishing Co Inc, 2000. doi: 10.1142/4016.

[14]

P. Kapitza, E7-Viscosity of liquid helium below the -point, Nature, 141 (1938), 74. doi: 10.1016/B978-0-08-015816-7.50011-0.

[15]

M. Kleman and O. D. Laverntovich, Soft Matter Physics: An Introduction, Springer Science Business Media, 2007. doi: 10.1007/b97416.

[16]

J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions in twodimensional systems, Journal of Physics C: Solid State Physics, 6 (1973), 1181. doi: 10.1088/0022-3719/6/7/010.

[17] L. D. Landau and E. M. Lifshitz, Statistical Physics: V. 5: Course of Theoretical Physics, Pergamon Press, 1969.
[18]

E. M. Lifschitz and L. P. Pitajewski, Lehrbuch Der Theoretischen Physik ("LandauLifschitz"). Band X, 2nd edition, Akademie-Verlag, Berlin, 1990. Physikalische Kinetik. [Physical kinetics], Translated from the Russian by Gerd Röpke and Thomas Frauenheim, Translation edited and with a foreword by Paul Ziesche and Gerhard Diener.

[19]

E. Lifshitz and L. Pitaevskii, Statistical Physics Part 2, Landau and Lifshitz Course of Theoretical Physics vol. 9, 1980.

[20] T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag,, New York, 2014. doi: 10.1007/978-1-4614-8963-4.
[21] T. Ma and S. Wang, Mathematical Principles of Theoretical Physics, Science Press, Beijing, 2015.
[22]

T. Ma and S. Wang, Dynamic law of physical motion and potential-descending principle, J. Math. Study, 50 (2017), 215-241. doi: 10.4208/jms.v50n3.17.02.

[23]

T. Ma and S. Wang, Statistical Theory of Heat, Hal preprint: Hal-01578634, 2017.

[24]

T. Ma and S. Wang, Topological phase transitions Ⅰ: Quantum phase transitions, to appear, (2018).

[25]

T. Ohmi and K. Machida, Bose-Einstein condensation with internal degrees of freedom in alkali atom gases, Journal of the Physical Society of Japan, 67 (1998), 1822-1825. doi: 10.1143/JPSJ.67.1822.

[26]

R. K. Pathria and P. D. Beale, Statistical Mechanics, 3nd edition, Elsevier, 2011.

[27]

L. Pitaevskii, Vortex lines in an imperfect bose gas, Sov. Phys. JETP, 13 (1961), 451-454.

[28]

L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation and Super Uidity, vol. 164 of Internat. Ser. Mono. Phys., Clarendon Press, Oxford, 2003.

[29]

L. E. Reichl, A modern Course in Statistical Physics, A Wiley-Interscience Publication, 2nd edition, John Wiley Sons Inc., New York, 1998.

[30]

H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, New York and Oxford, 1971.

[31]

M. Tinkham, Introduction to Superconductivity, McGraw-Hill, Inc, 1996.

Figure 3.1.  An electron rotating around a direction $n$ with velocity $v$ induces a magnetic moment $m = ev{\bf s}$, where ${\bf s}$ is the area vector enclosed by the electron orbit
Figure 4.1.  The coexistence curve of $^3$He without an applied magnetic field
Figure 4.2.  $PT$-phase diagram of $^3$He in a magnetic field
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