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December  2019, 11(4): 447-485. doi: 10.3934/jgm.2019023

Networks of coadjoint orbits: From geometric to statistical mechanics

Alexis Arnaudon and  So Takao ,

Department of Mathematics, Imperial College, London SW7 2AZ, UK

* Corresponding author: So Takao

To Darryl Holm, for his 70th birthday

Received  April 2018 Revised  May 2019 Published  November 2019

Fund Project: AA acknowledges EPSRC funding through award EP/N014529/1 via the EPSRC Centre for Mathematics of Precision Healthcare. ST acknowledges funding through Schrödinger scholarship scheme.

A class of network models with symmetry group $ G $ that evolve as a Lie-Poisson system is derived from the framework of geometric mechanics, which generalises the classical Heisenberg model studied in statistical mechanics. We considered two ways of coupling the spins: one via the momentum and the other via the position and studied in details the equilibrium solutions and their corresponding nonlinear stability properties using the energy-Casimir method. We then took the example $ G = SO(3) $ and saw that the momentum-coupled system reduces to the classical Heisenberg model with massive spins and the position-coupled case reduces to a new system that has a broken symmetry group $ SO(3)/SO(2) $ similar to the heavy top. In the latter system, we numerically observed an interesting synchronisation-like phenomenon for a certain class of initial conditions. Adding a type of noise and dissipation that preserves the coadjoint orbit of the network model, we found that the invariant measure is given by the Gibbs measure, from which the notion of temperature is defined. We then observed a surprising 'triple-humped' phase transition in the heavy top-like lattice model, where the spins switched from one equilibrium position to another before losing magnetisation as we increased the temperature. This work is only a first step towards connecting geometric mechanics with statistical mechanics and several interesting problems are open for further investigation.

Keywords: Euler-Poincaré reduction, semi-direct product reduction, lie groups, statistical mechanics, Langevin dynamics, phase transition.
Mathematics Subject Classification: Primary: 70E55, 82B20; Secondary: 37J25, 82B26.
Citation: Alexis Arnaudon, So Takao. Networks of coadjoint orbits: From geometric to statistical mechanics. Journal of Geometric Mechanics, 2019, 11 (4) : 447-485. doi: 10.3934/jgm.2019023
References:
[1]

M. Arnaudon, X. Chen and A. B. Cruzeiro, Stochastic Euler-Poincaré reduction, Journal of Mathematical Physics, 55 (2014), 081507, 17 pp. doi: 10.1063/1.4893357.  Google Scholar

[2]

A. Arnaudon, A. L. de Castro and D. D. Holm, Noise and dissipation in rigid body motion, Stochastic Geometric Mechanics, Springer Proc. Math. Stat., Springer, Cham, 202 (2017), 1-12.   Google Scholar

[3]

A. Arnaudon, A. L. De Castro and D. D. Holm, Noise and dissipation on coadjoint orbits, Journal of Nonlinear Science, 28 (2018), 91-145.  doi: 10.1007/s00332-017-9404-3.  Google Scholar

[4]

A. Arnaudon, N. Ganaba and D. D. Holm, The stochastic energy-Casimir method, Comptes Rendus Mécanique, 346 (2018), 279-290.  doi: 10.1016/j.crme.2018.01.003.  Google Scholar

[5]

A. Arnaudon, D. D. Holm and S. Sommer, A geometric framework for stochastic shape analysis, Foundation of Computational Mathematics, 19 (2017), 653–701, arXiv: 1703.09971. doi: 10.1007/s10208-018-9394-z.  Google Scholar

[6]

V. I. Arnol'd, An a priori estimate in the theory of hydrodynamic stability, Izvestiya Vysšhikh Učhebnykh Zavedenii. Matematika, 1996 (1996), 3-5.   Google Scholar

[7]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Second edition, Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[8]

M. Barahona and L. M. Pecora, Synchronization in small-world systems, Physical Review Letters, 89 (2002), 054101.  doi: 10.1103/PhysRevLett.89.054101.  Google Scholar

[9]

F. Barbaresco, Koszul information geometry and Souriau geometric temperature/capacity of Lie group thermodynamics, Entropy, 16 (2014), 4521-4565.  doi: 10.3390/e16084521.  Google Scholar

[10]

F. Barbaresco, Symplectic structure of information geometry: Fisher metric and Euler-Poincaré equation of Souriau Lie group thermodynamics, Geometric Science of Information, Lecture Notes in Comput. Sci., Springer, Cham, 9389 (2015), 529-540.  doi: 10.1007/978-3-319-25040-3_57.  Google Scholar

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F. Barbaresco, Geometric theory of heat from Souriau Lie groups thermodynamics and Koszul hessian geometry: Applications in information geometry for exponential families, Entropy, 16 (2014), 4521-4565.  doi: 10.3390/e16084521.  Google Scholar

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A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and T. S. Ratiu, Dissipation induced instabilities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 37-90.  doi: 10.1016/S0294-1449(16)30196-2.  Google Scholar

[14]

A. Bloch, P. S. Krishnaprasad, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and double bracket dissipation, Communications in Mathematical Physics, 175 (1996), 1-42.  doi: 10.1007/BF02101622.  Google Scholar

[15]

N. Bou-Rabee and H. Owhadi, Stochastic variational integrators, IMA J. Numer. Anal., 29 (2009), 421-443.  doi: 10.1093/imanum/drn018.  Google Scholar

[16] D. Chandler, Introduction to Modern Statistical Mechanics, The Clarendon Press, Oxford University Press, New York, 1987.   Google Scholar
[17]

H. Cendra, D. D. Holm, M. J. W. Hoyle and J. E. Marsden, The Maxwell-Vlasov equations in Euler-Poincaré form, Journal of Mathematical Physics, 39 (1998), 3188-3157.  doi: 10.1063/1.532244.  Google Scholar

[18]

P. Expert, S. de Nigris, T. Takaguchi and R. Lambiotte, Graph spectral characterization of the XY model on complex networks, Physical Review E, 96 (2017), 012312.  doi: 10.1103/PhysRevE.96.012312.  Google Scholar

[19]

F. Gay-Balmaz and D. D. Holm, Selective decay by Casimir dissipation in inviscid fluids, Nonlinearity, 26 (2013), 495-524.  doi: 10.1088/0951-7715/26/2/495.  Google Scholar

[20]

F. Gay-Balmaz and D. D. Holm, A geometric theory of selective decay with applications in MHD, Nonlinearity, 27 (2014), 1747-1777.  doi: 10.1088/0951-7715/27/8/1747.  Google Scholar

[21]

F. Gay-Balmaz and H. Yoshimura, A Lagrangian variational formulation for nonequilibrium thermodynamics. Part Ⅰ: Discrete systems, Journal of Geometry and Physics, 111 (2017), 169-193.  doi: 10.1016/j.geomphys.2016.08.018.  Google Scholar

[22]

F. Gay-Balmaz and H. Yoshimura, A Lagrangian variational formulation for nonequilibrium thermodynamics. Part Ⅱ: Continuum systems, Journal of Geometry and Physics, 111 (2017), 194-212.  doi: 10.1016/j.geomphys.2016.08.019.  Google Scholar

[23]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Advances in Mathematics, 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721.  Google Scholar

[24]

D. D. Holm, J. E. Marsden, T. Ratiu and A. Weinstein, Nonlinear stability of fluid and plasma equilibria, Physics Reports, 123 (1985), 116 pp. doi: 10.1016/0370-1573(85)90028-6.  Google Scholar

[25]

D. D. Holm, Geometric mechanics. Part II, Second edition, Imperial College Press, London, 2011. doi: 10.1142/p802.  Google Scholar

[26]

D. D. Holm, Variational principles for stochastic fluid dynamics, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 471 (2015), 20140963, 19 pp. doi: 10.1098/rspa.2014.0963.  Google Scholar

[27]

E. W. Justh and P. S. Krishnaprasad, Optimality, reduction and collective motion, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471 (2015), 20140606, 22 pp. doi: 10.1098/rspa.2014.0606.  Google Scholar

[28]

J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, Basic Notions Of Condensed Matter Physics, 24, 23 pp. doi: 10.4324/9780429494116-24.  Google Scholar

[29]

R. Kubo, The fluctuation-dissipation theorem, Reports on Progress in Physics, 29 (1966), 255.  doi: 10.1088/0034-4885/29/1/306.  Google Scholar

[30]

J.-A. Lázaro-Camí and J.-P. Ortega, Stochastic Hamiltonian dynamical systems, Rep. Math. Phys., 61 (2008), 65-122.  doi: 10.1016/S0034-4877(08)80003-1.  Google Scholar

[31]

C.-M. Marle, From tools in symplectic and poisson geometry to J.-M. Souriau's theories of statistical mechanics and thermodynamics, Entropy, 18 (2016), 370.  doi: 10.3390/e18100370.  Google Scholar

[32]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Second edition, Texts in Applied Mathematics, 17. Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[33]

T. Ratiu, Euler-Poisson equations on Lie algebras and the N-dimensional heavy rigid body, Proceedings of the National Academy of Sciences of the United States of America, 78 (1981), 1327-1328.  doi: 10.1073/pnas.78.3.1327.  Google Scholar

[34]

T. Ratiu and P. van Moerbeke, The Lagrange rigid body motion, Annales de l'institut Fourier, 32 (1982), 211-234.  doi: 10.5802/aif.866.  Google Scholar

[35]

D. Ruelle, Thermodynamic Formalism: The Mathematical Structure of Equilibrium Statistical Mechanics, Second edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511617546.  Google Scholar

[36]

J. M. Souriau, Definition covariante des equilibres thermodynamiques, Nuovo Cimento, 4 (1966), 203.   Google Scholar

[37]

J.-M. Souriau, Structure des Systemes Dynamiques, Maȋtrises de Mathématiques Dunod, Paris, 1970.  Google Scholar

[38]

J.-M. Souriau, Mécanique statistique, groupes de Lie et cosmologie, Géométrie Symplectique et physique Mathématique, Éditions Centre Nat. Recherche Sci., Paris, (1975), 59–113.  Google Scholar

show all references

To Darryl Holm, for his 70th birthday

References:
[1]

M. Arnaudon, X. Chen and A. B. Cruzeiro, Stochastic Euler-Poincaré reduction, Journal of Mathematical Physics, 55 (2014), 081507, 17 pp. doi: 10.1063/1.4893357.  Google Scholar

[2]

A. Arnaudon, A. L. de Castro and D. D. Holm, Noise and dissipation in rigid body motion, Stochastic Geometric Mechanics, Springer Proc. Math. Stat., Springer, Cham, 202 (2017), 1-12.   Google Scholar

[3]

A. Arnaudon, A. L. De Castro and D. D. Holm, Noise and dissipation on coadjoint orbits, Journal of Nonlinear Science, 28 (2018), 91-145.  doi: 10.1007/s00332-017-9404-3.  Google Scholar

[4]

A. Arnaudon, N. Ganaba and D. D. Holm, The stochastic energy-Casimir method, Comptes Rendus Mécanique, 346 (2018), 279-290.  doi: 10.1016/j.crme.2018.01.003.  Google Scholar

[5]

A. Arnaudon, D. D. Holm and S. Sommer, A geometric framework for stochastic shape analysis, Foundation of Computational Mathematics, 19 (2017), 653–701, arXiv: 1703.09971. doi: 10.1007/s10208-018-9394-z.  Google Scholar

[6]

V. I. Arnol'd, An a priori estimate in the theory of hydrodynamic stability, Izvestiya Vysšhikh Učhebnykh Zavedenii. Matematika, 1996 (1996), 3-5.   Google Scholar

[7]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Second edition, Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[8]

M. Barahona and L. M. Pecora, Synchronization in small-world systems, Physical Review Letters, 89 (2002), 054101.  doi: 10.1103/PhysRevLett.89.054101.  Google Scholar

[9]

F. Barbaresco, Koszul information geometry and Souriau geometric temperature/capacity of Lie group thermodynamics, Entropy, 16 (2014), 4521-4565.  doi: 10.3390/e16084521.  Google Scholar

[10]

F. Barbaresco, Symplectic structure of information geometry: Fisher metric and Euler-Poincaré equation of Souriau Lie group thermodynamics, Geometric Science of Information, Lecture Notes in Comput. Sci., Springer, Cham, 9389 (2015), 529-540.  doi: 10.1007/978-3-319-25040-3_57.  Google Scholar

[11]

F. Barbaresco, Geometric theory of heat from Souriau Lie groups thermodynamics and Koszul hessian geometry: Applications in information geometry for exponential families, Entropy, 16 (2014), 4521-4565.  doi: 10.3390/e16084521.  Google Scholar

[12]

J.-M. Bismut, Mécanique aléatoire, Tenth Saint Flour Probability Summer School—1980, Lecture Notes in Math., Springer, Berlin-New York, 929 (1982), 1–100  Google Scholar

[13]

A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and T. S. Ratiu, Dissipation induced instabilities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 37-90.  doi: 10.1016/S0294-1449(16)30196-2.  Google Scholar

[14]

A. Bloch, P. S. Krishnaprasad, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and double bracket dissipation, Communications in Mathematical Physics, 175 (1996), 1-42.  doi: 10.1007/BF02101622.  Google Scholar

[15]

N. Bou-Rabee and H. Owhadi, Stochastic variational integrators, IMA J. Numer. Anal., 29 (2009), 421-443.  doi: 10.1093/imanum/drn018.  Google Scholar

[16] D. Chandler, Introduction to Modern Statistical Mechanics, The Clarendon Press, Oxford University Press, New York, 1987.   Google Scholar
[17]

H. Cendra, D. D. Holm, M. J. W. Hoyle and J. E. Marsden, The Maxwell-Vlasov equations in Euler-Poincaré form, Journal of Mathematical Physics, 39 (1998), 3188-3157.  doi: 10.1063/1.532244.  Google Scholar

[18]

P. Expert, S. de Nigris, T. Takaguchi and R. Lambiotte, Graph spectral characterization of the XY model on complex networks, Physical Review E, 96 (2017), 012312.  doi: 10.1103/PhysRevE.96.012312.  Google Scholar

[19]

F. Gay-Balmaz and D. D. Holm, Selective decay by Casimir dissipation in inviscid fluids, Nonlinearity, 26 (2013), 495-524.  doi: 10.1088/0951-7715/26/2/495.  Google Scholar

[20]

F. Gay-Balmaz and D. D. Holm, A geometric theory of selective decay with applications in MHD, Nonlinearity, 27 (2014), 1747-1777.  doi: 10.1088/0951-7715/27/8/1747.  Google Scholar

[21]

F. Gay-Balmaz and H. Yoshimura, A Lagrangian variational formulation for nonequilibrium thermodynamics. Part Ⅰ: Discrete systems, Journal of Geometry and Physics, 111 (2017), 169-193.  doi: 10.1016/j.geomphys.2016.08.018.  Google Scholar

[22]

F. Gay-Balmaz and H. Yoshimura, A Lagrangian variational formulation for nonequilibrium thermodynamics. Part Ⅱ: Continuum systems, Journal of Geometry and Physics, 111 (2017), 194-212.  doi: 10.1016/j.geomphys.2016.08.019.  Google Scholar

[23]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Advances in Mathematics, 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721.  Google Scholar

[24]

D. D. Holm, J. E. Marsden, T. Ratiu and A. Weinstein, Nonlinear stability of fluid and plasma equilibria, Physics Reports, 123 (1985), 116 pp. doi: 10.1016/0370-1573(85)90028-6.  Google Scholar

[25]

D. D. Holm, Geometric mechanics. Part II, Second edition, Imperial College Press, London, 2011. doi: 10.1142/p802.  Google Scholar

[26]

D. D. Holm, Variational principles for stochastic fluid dynamics, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 471 (2015), 20140963, 19 pp. doi: 10.1098/rspa.2014.0963.  Google Scholar

[27]

E. W. Justh and P. S. Krishnaprasad, Optimality, reduction and collective motion, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471 (2015), 20140606, 22 pp. doi: 10.1098/rspa.2014.0606.  Google Scholar

[28]

J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, Basic Notions Of Condensed Matter Physics, 24, 23 pp. doi: 10.4324/9780429494116-24.  Google Scholar

[29]

R. Kubo, The fluctuation-dissipation theorem, Reports on Progress in Physics, 29 (1966), 255.  doi: 10.1088/0034-4885/29/1/306.  Google Scholar

[30]

J.-A. Lázaro-Camí and J.-P. Ortega, Stochastic Hamiltonian dynamical systems, Rep. Math. Phys., 61 (2008), 65-122.  doi: 10.1016/S0034-4877(08)80003-1.  Google Scholar

[31]

C.-M. Marle, From tools in symplectic and poisson geometry to J.-M. Souriau's theories of statistical mechanics and thermodynamics, Entropy, 18 (2016), 370.  doi: 10.3390/e18100370.  Google Scholar

[32]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Second edition, Texts in Applied Mathematics, 17. Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[33]

T. Ratiu, Euler-Poisson equations on Lie algebras and the N-dimensional heavy rigid body, Proceedings of the National Academy of Sciences of the United States of America, 78 (1981), 1327-1328.  doi: 10.1073/pnas.78.3.1327.  Google Scholar

[34]

T. Ratiu and P. van Moerbeke, The Lagrange rigid body motion, Annales de l'institut Fourier, 32 (1982), 211-234.  doi: 10.5802/aif.866.  Google Scholar

[35]

D. Ruelle, Thermodynamic Formalism: The Mathematical Structure of Equilibrium Statistical Mechanics, Second edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511617546.  Google Scholar

[36]

J. M. Souriau, Definition covariante des equilibres thermodynamiques, Nuovo Cimento, 4 (1966), 203.   Google Scholar

[37]

J.-M. Souriau, Structure des Systemes Dynamiques, Maȋtrises de Mathématiques Dunod, Paris, 1970.  Google Scholar

[38]

J.-M. Souriau, Mécanique statistique, groupes de Lie et cosmologie, Géométrie Symplectique et physique Mathématique, Éditions Centre Nat. Recherche Sci., Paris, (1975), 59–113.  Google Scholar

Figure 1.  In this figure, we plot the maximum real part of the eigenvalues $ \lambda_S $ of the linearised system (58) corresponding to equilibrium solutions $ {\boldsymbol \Pi}_e $ (eigenvectors of $ \mathbb L $) with energy $ \lambda_e $ (eigenvalue of $ \mathbb L $ corresponding to eigenvector $ {\boldsymbol \Pi}_e $). The blue crosses indicate the maximum real eigenvalues that are non-zero, corresponding to unstable solutions, and the red dots indicate those that are zero, corresponding to linearly stable solutions. The purple dots indicate the algebraic multiplicity $ n $ of $ \lambda_e $, whose values are to be read from the right axis. As expected, the equilibria corresponding to the highest and lowest energy are stable in both cases
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figure 1">Figure 2.  In this figure, we display four non-trivial examples of linearly stable equilibrium solutions of the $ 20\times 20 $ rigid body lattice. All the solutions are parallel to one of the axes (either $ \Pi_1 $ or $ \Pi_3 $) and have an 'argyle' pattern of aligned spins (2(a), 2(d)) or anti-aligned spins (2(b), 2(c)). The former pattern corresponds to configurations with low energy (negative values of $ \lambda $), and the latter corresponds to configurations with high energy (positive values of $ \lambda $). All four solutions given here correspond to the red dots in figure 1
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Figure 3.  This figure shows the possible $ (\lambda_1, \lambda_2) $-pairs that one can have from solving equation (86). Fixing $ \lambda_1 $, the possible values of $ \lambda_2 $ that solve (86) are the eigenvalues of the graph Laplacian $ \mathbb L(\lambda_1) $. We see that the eigenvalues $ \lambda_2 $ in the left panel 3(a) collapse as $ \lambda_1 \rightarrow 0 $, which does not occur in the right panel 3(b)
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Figure 4.  The two panels in this figure display the maximum, real part of the eigenvalues $ \lambda_S $ of the linearised system (98) around the equilibrium configuration $ {\boldsymbol \Gamma}_e $ (eigenvectors of $ \mathbb L(\lambda_1) $) plotted against $ \lambda_2 $ (minus the eigenvalue of $ \mathbb L(\lambda_1) $ corresponding to $ {\boldsymbol \Gamma}_e $) for $ \lambda_1 = 0.5 $. Again, the blue crosses correspond to unstable equilibria, the red dots correspond to linearly stable equilibria and the purple dots correspond to the multiplicity of $ \lambda_2 $, whose values are to be read from the right axis. As expected, the configuration corresponding to the highest $ \lambda_2 $ (or, the lowest eigenvalue of $ \mathbb L(\lambda_1) $) is stable in both cases
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Figure 5.  This figure shows the time evolution of the averaged spins (5(a), 5(c), 5(e)) and energy (5(b), 5(d), 5(f)) in a $ 20\times 20 $ heavy top lattice, starting from different initial conditions, with dissipation (dashed lines) or without dissipation (bold lines). The blue, orange and green lines in the left panel correspond to the $ \Gamma_1 $, $ \Gamma_2 $ and $ \Gamma_3 $-components of the spins averaged over the lattice respectively and the same colours in the right panel correspond to the total, kinetic and potential energy. We observe metastability of the $ \Gamma_1 $ and $ \Gamma_2 $ axes in panels 5(c) and 5(a) in the dissipative case, and a partial synchronisation around the $ \Gamma_3 $-axis in the non-dissipative case in 5(a), 5(c) and 5(e)
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Figure 6.  The two panels in this figure show the temperature phase transition of the rigid body lattice for two different cases. The blue dots correspond to averaged magnetisation computed via direct simulation of the full stochastic equation (47) and the black line corresponds to the mean field approximation (103). Both cases exhibit a second order phase transition, characterised by a sudden loss of magnetisation (magnitude of the order parameter) at the critical temperature
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figure 7(a), the blue dots represent the $ \Gamma_3 $-components of the averaged position $ \left\langle {\bf{\Gamma }} \right\rangle $ and in 7(b)-7(d), the blue, green and red dots represent the $ \Gamma_1 $, $ \Gamma_2 $ and $ \Gamma_3 $-components of the averaged position respectively. The black line in all four figures correspond to the mean field approximation (106). The top left panel 7(a) shows a phase transition similar to the rigid body phase transition, and the other panels 7(b)-7(d) display a 'triple-humped' phase transition. The corresponding critical temperatures are indicated by the vertical dashed lines. In 7(b)-7(d), we also varied the Casimirs (where $ c_1 = {\boldsymbol \Pi} \cdot {\boldsymbol \Gamma} $ and $ c_1 = \|{\boldsymbol \Gamma}\|^2 $) to show that the three critical temperatures depend on their values, and may even disappear for large enough $ c_1/c_2 $. Due to the high dimension and non-compactness of the phase space, the mean field approximation requires intensive computation to be precise, explaining the visible errors, especially at low temperatures">Figure 7.  The four panels in this figure show phase transitions in the heavy top lattice. In figure 7(a), the blue dots represent the $ \Gamma_3 $-components of the averaged position $ \left\langle {\bf{\Gamma }} \right\rangle $ and in 7(b)-7(d), the blue, green and red dots represent the $ \Gamma_1 $, $ \Gamma_2 $ and $ \Gamma_3 $-components of the averaged position respectively. The black line in all four figures correspond to the mean field approximation (106). The top left panel 7(a) shows a phase transition similar to the rigid body phase transition, and the other panels 7(b)-7(d) display a 'triple-humped' phase transition. The corresponding critical temperatures are indicated by the vertical dashed lines. In 7(b)-7(d), we also varied the Casimirs (where $ c_1 = {\boldsymbol \Pi} \cdot {\boldsymbol \Gamma} $ and $ c_1 = \|{\boldsymbol \Gamma}\|^2 $) to show that the three critical temperatures depend on their values, and may even disappear for large enough $ c_1/c_2 $. Due to the high dimension and non-compactness of the phase space, the mean field approximation requires intensive computation to be precise, explaining the visible errors, especially at low temperatures
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