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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

* Corresponding author: Dongpei Zhang

Received  September 2018 Published  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.

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