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September
2019, 24(9): 5107-5120.
doi: 10.3934/dcdsb.2019045
Superfluidity phase transitions for liquid $ ^{4} $He system
| 1. | School of Medical Informatics and Engineering, Southwest Medical University, Luzhou, Sichuan 646000, China |
| 2. | Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China |
The main objective of this paper is to investigate the superfluidity phase transition theory-modeling and analysis-for liquid $ ^{4} $He system. Based on the new Gibbs free energy and the potential-descending principle proposed recently in [
Mathematics Subject Classification:
Primary: 34C23, 35Q99, 37G35; Secondary: 37L10.
Citation:
Jiayan Yang, Dongpei Zhang. Superfluidity phase transitions for liquid $ ^{4} $He system. Discrete and Continuous Dynamical Systems - B,
2019, 24
(9)
: 5107-5120.
doi: 10.3934/dcdsb.2019045
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References:
| [1] |
J. F. Allen and A. D. Misener, Flow of liquid Helium Ⅱ, Nature, 141 (1938), 75.
doi: 10.1038/141075a0. |
| [2] |
A. Berti and V. Berti,
A thermodynamically consistent Ginzburg-Landau model for superfluid transition in liquid helium, Z. Angew. Math. Phys., 64 (2013), 1387-1397.
doi: 10.1007/s00033-012-0280-2. |
| [3] |
A. Berti, V. Berti and I. Bochicchio,
Global and exponential attractors for a Ginzburg-Landau model of superfluidity, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 247-271.
doi: 10.3934/dcdss.2011.4.247. |
| [4] |
A. Berti, V. Berti and I. Bochicchio, Asymptotic behavior of Ginzburg-Landau equations of superfluidity, Communications to SIMAI Congress, 3 (2009), 12pp. |
| [5] |
V. Berti and M. Fabrizio, Well-posedness for a Ginzburg-Landau model in superfluidity, in New Trends in Fluid and Solid Models, World Scientific, (2009), 1–9.
doi: 10.1142/9789814293228_0001. |
| [6] |
V. Berti and M. Fabrizio,
Existence and uniqueness for a mathematical model in superfluidity, Math. Meth. Appl. Sci., 31 (2008), 1441-1459.
doi: 10.1002/mma.981. |
| [7] |
M. Campostrini, M. Hasenbusch, A. Pelissetto and E. Vicari,
Theoretical estimates of the critical exponents of the superfluid transition in $^4$He by lattice methods, Phys. Rev. B, 74 (2006), 2952-2961.
|
| [8] |
A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzsber. Preuss. Akad., 3 (2006).
doi: 10.1002/3527608958.ch27. |
| [9] |
M. Fabrizio,
A Ginzburg-Landau model for the phase transition in Helium Ⅱ, Z. Angew. Math. Phys., 61 (2010), 329-340.
doi: 10.1007/s00033-009-0011-5. |
| [10] |
M. Fabrizio and M. S. Mongiovì,
Phase transition in liquid $^4$He by a mean field model, J. Therm. Stresses, 36 (2013), 135-151.
|
| [11] |
H. A. Gersch and J. M. Tanner,
Solid-superfluid transition in $^4$He at absolute zero, Phys. Rev., 139 (1965), 1769-1782.
|
| [12] |
V. L. Ginzburg and L. D. Landau,
On the theory of superconductivity, Zh. Eksp. Teor. Fiz., 20 (1950), 1064-1082.
doi: 10.1007/978-3-540-68008-6_4. |
| [13] |
Z. B. Hou and L. M. Li,
Global attractor of the liquid Helium-4 system in $H^{k}$ space, Appl. Mech. Mater., 444/445 (2014), 731-737.
doi: 10.4028/www.scientific.net/AMM.444-445.731. |
| [14] |
P. Kapitza, Viscosity of liquid Helium below the $\lambda$-point, Nature, 141 (1938), 74. |
| [15] |
S. Koh, Shear viscosity of liquid helium 4 above the lambda point, Physics, 2008. |
| [16] |
L. D. Landau, Theory of superfluidity of Helium-Ⅱ, Zh. Eksp. Teor. Fiz., 11 (1941). |
| [17] |
T. Lindenau, M. L. Ristig, J. W. Clark and K. A. Gernoth,
Bose-Einstein condensation and the $\lambda$-transition in liquid Helium, J. Low Temp. Phys., 129 (2002), 143-170.
|
| [18] |
R. K. Liu, T. Ma, S. H. Wang and J. Y. Yang, Thermodynamical potentials of classical and quantum systems, Discrete Contin. Dyn. Syst. Ser. B, (2018) (to appear).
doi: 10.3934/dcdsb.2018214. |
| [19] |
F. London,
The $\lambda$-phenomenon of liquid Helium and the Bose-Einstein degeneracy, Nature, 141 (1938), 643-644.
|
| [20] |
T. Ma, R. K. Liu and J. Y. Yang, Physical World from the Mathematical Point of View: Statistical Physics and Critical Phase Transition Theory(in Chinese), Science Press, Beijing, 2017. |
| [21] |
T. Ma and S. H. Wang, Bifurcation Theory and Applications, World Scientific Publishing Co. Pte. Ltd.: Hackensack, NJ, 2005.
doi: 10.1142/5827. |
| [22] |
T. Ma and S. H. Wang, Phase Transition Dynamics, Springer-Verlag, New York, 2014.
doi: 10.1007/978-1-4614-8963-4. |
| [23] |
T. Ma and S. H. Wang, Dynamic model and phase transitions for liquid helium, J. Math. Phys., 49 (2008), 073304, 18 pp.
doi: 10.1063/1.2957943. |
| [24] |
T. Ma and S. H. Wang, Phase transition and separation for mixture of liquid He-3 and He-4, in Lev Davidovich Landau and his Impact on Contemporary Theoretical Physics, Nova Science Publishers Inc; UK ed., (2010), 107–119. |
| [25] |
T. Ma and S. H. Wang,
Dynamic law of physical motion and potential-descending principle, J. Math. Study, 50 (2017), 215-241.
doi: 10.4208/jms.v50n3.17.02. |
| [26] |
V. N. Minasyan and V. N. Samoilov,
The condition of existence of the Bose-Einstein condensation in the superfluid liquid helium, Phys. Lett. A, 374 (2010), 2792-2797.
doi: 10.1016/j.physleta.2010.04.072. |
| [27] |
M. S. Mongiovì and L. Saluto,
Effects of heat flux on $\lambda$-transition in liquid $^4$He, Meccanica, 49 (2014), 2125-2137.
doi: 10.1007/s11012-014-9922-0. |
| [28] |
O. Penrose and L. Onsager,
Bose-Einstein condensation and liquid Helium, Phys. Rev., 104 (1956), 576-584.
doi: 10.4324/9780429494116-14. |
| [29] |
J. K. Perron, M. O. Kimball, K. P. Mooney and F. M. Gasparini, Critical behavior of coupled $^4$He regions near the superfluid transition, Phys. Rev. B, 87 (2013), 094507. |
| [30] |
L. Tisza, Transport phenomena in Helium Ⅱ, Nature, 141 (1938), 913.
doi: 10.1038/141913a0. |
show all references
References:
| [1] |
J. F. Allen and A. D. Misener, Flow of liquid Helium Ⅱ, Nature, 141 (1938), 75.
doi: 10.1038/141075a0. |
| [2] |
A. Berti and V. Berti,
A thermodynamically consistent Ginzburg-Landau model for superfluid transition in liquid helium, Z. Angew. Math. Phys., 64 (2013), 1387-1397.
doi: 10.1007/s00033-012-0280-2. |
| [3] |
A. Berti, V. Berti and I. Bochicchio,
Global and exponential attractors for a Ginzburg-Landau model of superfluidity, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 247-271.
doi: 10.3934/dcdss.2011.4.247. |
| [4] |
A. Berti, V. Berti and I. Bochicchio, Asymptotic behavior of Ginzburg-Landau equations of superfluidity, Communications to SIMAI Congress, 3 (2009), 12pp. |
| [5] |
V. Berti and M. Fabrizio, Well-posedness for a Ginzburg-Landau model in superfluidity, in New Trends in Fluid and Solid Models, World Scientific, (2009), 1–9.
doi: 10.1142/9789814293228_0001. |
| [6] |
V. Berti and M. Fabrizio,
Existence and uniqueness for a mathematical model in superfluidity, Math. Meth. Appl. Sci., 31 (2008), 1441-1459.
doi: 10.1002/mma.981. |
| [7] |
M. Campostrini, M. Hasenbusch, A. Pelissetto and E. Vicari,
Theoretical estimates of the critical exponents of the superfluid transition in $^4$He by lattice methods, Phys. Rev. B, 74 (2006), 2952-2961.
|
| [8] |
A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzsber. Preuss. Akad., 3 (2006).
doi: 10.1002/3527608958.ch27. |
| [9] |
M. Fabrizio,
A Ginzburg-Landau model for the phase transition in Helium Ⅱ, Z. Angew. Math. Phys., 61 (2010), 329-340.
doi: 10.1007/s00033-009-0011-5. |
| [10] |
M. Fabrizio and M. S. Mongiovì,
Phase transition in liquid $^4$He by a mean field model, J. Therm. Stresses, 36 (2013), 135-151.
|
| [11] |
H. A. Gersch and J. M. Tanner,
Solid-superfluid transition in $^4$He at absolute zero, Phys. Rev., 139 (1965), 1769-1782.
|
| [12] |
V. L. Ginzburg and L. D. Landau,
On the theory of superconductivity, Zh. Eksp. Teor. Fiz., 20 (1950), 1064-1082.
doi: 10.1007/978-3-540-68008-6_4. |
| [13] |
Z. B. Hou and L. M. Li,
Global attractor of the liquid Helium-4 system in $H^{k}$ space, Appl. Mech. Mater., 444/445 (2014), 731-737.
doi: 10.4028/www.scientific.net/AMM.444-445.731. |
| [14] |
P. Kapitza, Viscosity of liquid Helium below the $\lambda$-point, Nature, 141 (1938), 74. |
| [15] |
S. Koh, Shear viscosity of liquid helium 4 above the lambda point, Physics, 2008. |
| [16] |
L. D. Landau, Theory of superfluidity of Helium-Ⅱ, Zh. Eksp. Teor. Fiz., 11 (1941). |
| [17] |
T. Lindenau, M. L. Ristig, J. W. Clark and K. A. Gernoth,
Bose-Einstein condensation and the $\lambda$-transition in liquid Helium, J. Low Temp. Phys., 129 (2002), 143-170.
|
| [18] |
R. K. Liu, T. Ma, S. H. Wang and J. Y. Yang, Thermodynamical potentials of classical and quantum systems, Discrete Contin. Dyn. Syst. Ser. B, (2018) (to appear).
doi: 10.3934/dcdsb.2018214. |
| [19] |
F. London,
The $\lambda$-phenomenon of liquid Helium and the Bose-Einstein degeneracy, Nature, 141 (1938), 643-644.
|
| [20] |
T. Ma, R. K. Liu and J. Y. Yang, Physical World from the Mathematical Point of View: Statistical Physics and Critical Phase Transition Theory(in Chinese), Science Press, Beijing, 2017. |
| [21] |
T. Ma and S. H. Wang, Bifurcation Theory and Applications, World Scientific Publishing Co. Pte. Ltd.: Hackensack, NJ, 2005.
doi: 10.1142/5827. |
| [22] |
T. Ma and S. H. Wang, Phase Transition Dynamics, Springer-Verlag, New York, 2014.
doi: 10.1007/978-1-4614-8963-4. |
| [23] |
T. Ma and S. H. Wang, Dynamic model and phase transitions for liquid helium, J. Math. Phys., 49 (2008), 073304, 18 pp.
doi: 10.1063/1.2957943. |
| [24] |
T. Ma and S. H. Wang, Phase transition and separation for mixture of liquid He-3 and He-4, in Lev Davidovich Landau and his Impact on Contemporary Theoretical Physics, Nova Science Publishers Inc; UK ed., (2010), 107–119. |
| [25] |
T. Ma and S. H. Wang,
Dynamic law of physical motion and potential-descending principle, J. Math. Study, 50 (2017), 215-241.
doi: 10.4208/jms.v50n3.17.02. |
| [26] |
V. N. Minasyan and V. N. Samoilov,
The condition of existence of the Bose-Einstein condensation in the superfluid liquid helium, Phys. Lett. A, 374 (2010), 2792-2797.
doi: 10.1016/j.physleta.2010.04.072. |
| [27] |
M. S. Mongiovì and L. Saluto,
Effects of heat flux on $\lambda$-transition in liquid $^4$He, Meccanica, 49 (2014), 2125-2137.
doi: 10.1007/s11012-014-9922-0. |
| [28] |
O. Penrose and L. Onsager,
Bose-Einstein condensation and liquid Helium, Phys. Rev., 104 (1956), 576-584.
doi: 10.4324/9780429494116-14. |
| [29] |
J. K. Perron, M. O. Kimball, K. P. Mooney and F. M. Gasparini, Critical behavior of coupled $^4$He regions near the superfluid transition, Phys. Rev. B, 87 (2013), 094507. |
| [30] |
L. Tisza, Transport phenomena in Helium Ⅱ, Nature, 141 (1938), 913.
doi: 10.1038/141913a0. |
Figure 2.
The diagram for critical control parameters $ (T_c, p_c) $
Figure 4.
The topology of steady state solutions for (24)
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