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September  2019, 24(9): 5107-5120. doi: 10.3934/dcdsb.2019045

Superfluidity phase transitions for liquid $ ^{4} $He system

Jiayan Yang 1, and  Dongpei Zhang 2,,

1. 

School of Medical Informatics and Engineering, Southwest Medical University, Luzhou, Sichuan 646000, China
2. 

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

* Corresponding author: Dongpei Zhang

Received  September 2018 Published  September 2019 Early access  February 2019

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 [18,25], the dynamic equations describing the $ \lambda $-transition and solid-liquid transition of liquid $ ^{4} $He system are derived. Further, by the dynamical transition theory, the two obtained models are proven to exhibit Ehrenfest second-order transition and first-order transition, respectively, which are well consistent with the physical experimental results.

Keywords: $ ^{4} $He system, superfluidity, Gibbs free energy, phase transition.
Mathematics Subject Classification: Primary: 34C23, 35Q99, 37G35; Secondary: 37L10.
Citation: Jiayan Yang, Dongpei Zhang. Superfluidity phase transitions for liquid $ ^{4} $He system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5107-5120. doi: 10.3934/dcdsb.2019045
References:
[1]

J. F. Allen and A. D. Misener, Flow of liquid Helium Ⅱ, Nature, 141 (1938), 75. doi: 10.1038/141075a0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

A. Berti, V. Berti and I. Bochicchio, Asymptotic behavior of Ginzburg-Landau equations of superfluidity, Communications to SIMAI Congress, 3 (2009), 12pp. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[8]

A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzsber. Preuss. Akad., 3 (2006). doi: 10.1002/3527608958.ch27.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[11]

H. A. Gersch and J. M. Tanner, Solid-superfluid transition in $^4$He at absolute zero, Phys. Rev., 139 (1965), 1769-1782.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

P. Kapitza, Viscosity of liquid Helium below the $\lambda$-point, Nature, 141 (1938), 74. Google Scholar

[15]

S. Koh, Shear viscosity of liquid helium 4 above the lambda point, Physics, 2008. Google Scholar

[16]

L. D. Landau, Theory of superfluidity of Helium-Ⅱ, Zh. Eksp. Teor. Fiz., 11 (1941). Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[19]

F. London, The $\lambda$-phenomenon of liquid Helium and the Bose-Einstein degeneracy, Nature, 141 (1938), 643-644.   Google Scholar

[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. Google Scholar

[21]

T. Ma and S. H. Wang, Bifurcation Theory and Applications, World Scientific Publishing Co. Pte. Ltd.: Hackensack, NJ, 2005. doi: 10.1142/5827.  Google Scholar

[22]

T. Ma and S. H. Wang, Phase Transition Dynamics, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-8963-4.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

O. Penrose and L. Onsager, Bose-Einstein condensation and liquid Helium, Phys. Rev., 104 (1956), 576-584.  doi: 10.4324/9780429494116-14.  Google Scholar

[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. Google Scholar

[30]

L. Tisza, Transport phenomena in Helium Ⅱ, Nature, 141 (1938), 913. doi: 10.1038/141913a0.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

A. Berti, V. Berti and I. Bochicchio, Asymptotic behavior of Ginzburg-Landau equations of superfluidity, Communications to SIMAI Congress, 3 (2009), 12pp. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[8]

A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzsber. Preuss. Akad., 3 (2006). doi: 10.1002/3527608958.ch27.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[11]

H. A. Gersch and J. M. Tanner, Solid-superfluid transition in $^4$He at absolute zero, Phys. Rev., 139 (1965), 1769-1782.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

P. Kapitza, Viscosity of liquid Helium below the $\lambda$-point, Nature, 141 (1938), 74. Google Scholar

[15]

S. Koh, Shear viscosity of liquid helium 4 above the lambda point, Physics, 2008. Google Scholar

[16]

L. D. Landau, Theory of superfluidity of Helium-Ⅱ, Zh. Eksp. Teor. Fiz., 11 (1941). Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[19]

F. London, The $\lambda$-phenomenon of liquid Helium and the Bose-Einstein degeneracy, Nature, 141 (1938), 643-644.   Google Scholar

[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. Google Scholar

[21]

T. Ma and S. H. Wang, Bifurcation Theory and Applications, World Scientific Publishing Co. Pte. Ltd.: Hackensack, NJ, 2005. doi: 10.1142/5827.  Google Scholar

[22]

T. Ma and S. H. Wang, Phase Transition Dynamics, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-8963-4.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

O. Penrose and L. Onsager, Bose-Einstein condensation and liquid Helium, Phys. Rev., 104 (1956), 576-584.  doi: 10.4324/9780429494116-14.  Google Scholar

[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. Google Scholar

[30]

L. Tisza, Transport phenomena in Helium Ⅱ, Nature, 141 (1938), 913. doi: 10.1038/141913a0.  Google Scholar

Figure 1.  $ T $-$ p $ phase diagram of $ ^{4} $He
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Figure 2.  The diagram for critical control parameters $ (T_c, p_c) $
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Figure 3.  The theoretical $ T $-$ p $ phase diagram for $ ^{4} $He
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Figure 4.  The topology of steady state solutions for (24)
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Figure 5.  The $ p $-$ \rho_n $ dynamical phase diagram
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