Abstract

This concise review is aimed at providing an introduction to the kinetic theory of partially coherent optical waves propagating in nonlinear media. The subject of incoherent nonlinear optics received a renewed interest since the first experimental demonstration of incoherent solitons in slowly responding photorefractive crystals. Several theories have been successfully developed to provide a detailed description of the novel dynamical features inherent to partially coherent nonlinear optical waves. However, such theories leave unanswered the following important question: Which is the long term (spatiotemporal) evolution of a partially incoherent optical field propagating in a nonlinear medium? In complete analogy with kinetic gas theory, one may expect that the incoherent field may evolve, owing to nonlinearity, towards a thermodynamic equilibrium state. Weak-turbulence theory is shown to describe the essential properties of this irreversible process of thermal wave relaxation to equilibrium. Precisely, the theory describes an irreversible evolution of the spectrum of the field towards a thermodynamic equilibrium state. The irreversible behavior is expressed through the H-theorem of entropy growth, whose origin is analogous to the celebrated Boltzmann’s H-theorem of kinetic gas theory. It is shown that thermal wave relaxation to equilibrium may be characterized by the existence of a genuine condensation process, whose thermodynamic properties are analogous to those of Bose-Einstein condensation, despite the fact that the considered optical wave is completely classical. In spite of the formal reversibility of optical wave propagation, the condensation process occurs by means of an irreversible evolution of the field towards a homogeneous plane-wave (condensate) with small-scale fluctuations superimposed (uncondensed particles), which store the information necessary for the reversible propagation. As a remarkable result, an increase of entropy (“disorder”) in the optical field requires the generation of a coherent structure (plane-wave). We show that, beyond the standard thermodynamic limit, wave condensation also occurs in two spatial dimensions. The numerical simulations are in quantitative agreement with the kinetic wave theory, without any adjustable parameter.

© 2007 Optical Society of America

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2006 (7)

A. Picozzi, "Nonequilibrated Oscillations of Coherence in Coupled Nonlinear Wave Systems," Phys. Rev. Lett. 96, 013905 (2006).
[CrossRef] [PubMed]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, "Velocity Locking of Incoherent Nonlinear Wave Packets," Phys. Rev. Lett. 97, 033902 (2006).
[CrossRef] [PubMed]

O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, "Incoherent solitons in instantaneous nonlocal nonlinear media," Phys. Rev. E 73, 015601 (2006).
[CrossRef]

O. Jedrkiewicz, A. Picozzi, M. Clerici, D. Faccio, and P. Di Trapani, "Emergence of X-Shaped Spatiotemporal Coherence in Optical Waves," Phys. Rev. Lett. 97, 243903 (2006).
[CrossRef]

S. Nazarenko and M. Onorato "Wave turbulence and vortices in Bose-Einstein condensation," Physica D 219, 1 (2006).
[CrossRef]

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, "Observation of attraction between Dark Solitons," Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

2005 (7)

A. Picozzi and P. Aschieri, "Influence of dispersion on the resonant interaction between three incoherent waves," Phys. Rev. E 72, 046606 (2005).
[CrossRef]

Ting-Sen Ku, Ming-Feng Shih, A. A. Sukhorukov, and Y. S. Kivshar "Coherence controlled soliton interactions,"Phys. Rev. Lett. 94, 063904 (2005).
[CrossRef] [PubMed]

V. E. Zakharov, S. V. Nazarenko, "Dynamics of the Bose-Einstein condensation," Physica D 201, 203-211 (2005).
[CrossRef]

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, D. N. Christodoulides, "Observation of random-phase lattice solitons," Nature (London) 433, 500 (2005).
[CrossRef] [PubMed]

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, "Condensation of Classical Nonlinear Waves," Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

K. G. Makris, H. Sarkissian, D. N. Christodoulides, and G. Assanto "Nonlocal incoherent spatial solitons in liquid crystals," J. Opt. Soc. Am. B 22, 1371-1377 (2005).
[CrossRef]

A. Sauter, S. Pitois, G. Millot, A. Picozzi, "Incoherent modulation instability in instantaneous nonlinear Kerr media," Opt. Lett. 30, 2143-2145 (2005).
[CrossRef] [PubMed]

2004 (12)

C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar "Partially Incoherent Optical Vortices in Self-Focusing Nonlinear Media," Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

K. Motzek, F. Kaiser, J. R. Salgueiro, Y. Kivshar, and C. Denz, "Incoherent vector vortex-mode solitons in selffocusing nonlinear media," Opt. Lett. 29, 2285 (2004).
[CrossRef] [PubMed]

H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides "Random-Phase Solitons in Nonlinear Periodic Lattices," Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef] [PubMed]

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, "Incoherent Solitons in Instantaneous Response Nonlinear Media," Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef] [PubMed]

A. Picozzi and M. Haelterman, "Condensation in Hamiltonian Parametric Wave Interaction," Phys. Rev. Lett. 92, 103901 (2004).
[CrossRef] [PubMed]

M. Kolesik, E. M. Wright, and J. V. Moloney, "Dynamic Nonlinear XWaves for Femtosecond Pulse Propagation in Water," Phys. Rev. Lett. 92, 253901 (2004).
[CrossRef] [PubMed]

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, "Features of modulational instability of partially coherent light: Importance of the incoherence spectrum," Phys. Rev. E 69, 025601 (2004).
[CrossRef]

A. Mussot, E. Lantz, H. Maillotte, T. Sylvestre, C. Finot, and S. Pitois, "Spectral broadening of a partially coherent CW laser beam in single-mode optical fibers," Opt. Exp. 12, 2838 (2004).
[CrossRef]

R. Y. Chiao, T. H. Hansson, J. M. Leinaas, and S. Viefers, "Effective photon-photon interaction in a twodimensional photon fluid," Phys. Rev. A 69, 063816 (2004).
[CrossRef]

S. A. Ponomarenko, N. M. Litchinitser, and G. P. Agrawal "Theory of incoherent optical solitons: Beyond the mean-field approximation," Phys. Rev. E 70, 015603 (2004).
[CrossRef]

S. A. Ponomarenko and G. P. Agrawal "Asymmetric incoherent vector solitons," Phys. Rev. E 69, 036604 (2004).
[CrossRef]

C. Montes, A. Picozzi and K. Gallo, "Ultra-coherent signal output from an incoherent cw-pumped singly resonant optical parametric oscillator," Opt. Comm. 237, 437-449 (2004).
[CrossRef]

2003 (6)

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, "Nonlinear Electromagnetic X-Waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated x-shaped light bullets," Phys. Rev. Lett. 91, 093904 (2003);
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L. Helczynski, M. Lisak, and D. Anderson "Influence of higher-order dispersion on modulational instability and pulse broadening of partially incoherent light," Phys. Rev. E 67, 026602 (2003).
[CrossRef]

H. Buljan, M. Soljacic, T. Carmon, and M. Segev, "Cavity pattern formation with incoherent light," Phys. Rev. E 68, 016616 (2003).
[CrossRef]

M. Lisak, L. Helczynski, D. Anderson, "Relation between different formalisms describing partially incoherent wave propagation in nonlinear optical media," Opt. Commun. 220, 321 (2003).
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J. Garnier, J.-P. Ayanides, and O. Morice, "Propagation of partially coherent light with the Maxwell-Debye equation," J. Opt. Soc. Am. B 20, 1409-1417 (2003).
[CrossRef]

2002 (6)

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov "Statistical theory for incoherent light propagation in nonlinear media," Phys. Rev. E 65, 035602 (2002).
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S. M. Sears, M. Soljacic, D. N. Christodoulides, and M. Segev, "Pattern formation via symmetry breaking in nonlinear weakly correlated systems," Phys. Rev. E 65, 036620 (2002).
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A. Picozzi and M. Haelterman "Hidden Coherence Along Space-Time Trajectories in Parametric Wave Mixing," Phys. Rev. Lett. 88, 083901 (2002).
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M. J. Davis, S. A. Morgan, and K. Burnett, "Simulations of thermal Bose fields in the classical limit," Phys. Rev. A 66, 053618 (2002).
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A. Picozzi, C. Montes, and M. Haelterman, "Coherence properties of the parametric three-wave interaction driven from an incoherent pump," Phys. Rev. E 66, 056605 (2002).
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N. G. Berloff and B. V. Svistunov "Scenario of strongly nonequilibrated Bose-Einstein condensation," Phys. Rev. A 66, 013603 (2002).
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2001 (7)

B. Rumpf and A. C. Newell, "Coherent structures and entropy in constrained, modulationally unstable nonintegrable systems," Phys. Rev. Lett. 87, 054102 (2001).
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R. Lacaze, P. Lallemand, Y. Pomeau and S. Rica, "Dynamical formation of a BoseEinstein condensate," Physica D 152-153, 779-786 (2001).
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M. J. Davis, S. A. Morgan and K. Burnett, "Simulations of Bose Fields at Finite Temperature," Phys. Rev. Lett. 87, 160402 (2001).
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A. Picozzi and M. Haelterman, "Parametric Three-Wave Soliton Generated from Incoherent Light," Phys. Rev. Lett. 86, 2010-2013 (2001).
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D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601 (2001).
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A. C. Newell, S. Nazarenko and L. Biven, "Wave turbulence and intermittency," Physica D 152-153, 520-550 (2001).
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M. Peccianti and G. Assanto "Incoherent spatial solitary waves in nematic liquid crystals," Opt. Lett. 26, 1791- 1793 (2001).
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2000 (6)

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath "Modulation Instability of Incoherent Beams in Noninstantaneous Nonlinear Media," Phys. Rev. Lett. 84, 467 (2000).
[CrossRef] [PubMed]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams," Science 290, 495-498 (2000).
[CrossRef] [PubMed]

C. Anastassiou, M. Soljacic, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, "Eliminating the Transverse Instabilities of Kerr Solitons," Phys. Rev. Lett. 85, 4888 (2000).
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R. Jordan and C. Josserand, "Self-organization in nonlinear wave turbulence," Phys. Rev. E 61, 1527-1539 (2000).
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W. Krolikowski, D. Edmundson, and O. Bang, "Unified model for partially coherent solitons in logarithmically nonlinear media," Phys. Rev. E 61, 3122 (2000).
[CrossRef]

T. H. Coskun, A. G. Grandpierre, D. N. Christodoulides, and M. Segev "Coherence enhancement of spatially incoherent light beams through soliton interactions," Opt. Lett. 25, 826 (2000).
[CrossRef]

1999 (4)

A. A. Sukhorukov and N. N. Akhmediev "Coherent and Incoherent Contributions to Multisoliton Complexes," Phys. Rev. Lett. 83, 4736 (1999).
[CrossRef]

O. Bang, D. Edmundson, andW. Krolikowski, "Collapse of Incoherent Light Beams in Inertial Bulk Kerr Media," Phys. Rev. Lett. 83, 5479 (1999).
[CrossRef]

T. H. Coskun, D. N. Christodoulides, Z. Chen, and M. Segev "Dark incoherent soliton splitting and phasememory effects: Theory and experiment," Phys. Rev. E 59, R4777 (1999).
[CrossRef]

R. Y. Chiao and J. Boyce "Bogoliubov dispersion relation and the possibility of superfluidity for weakly interacting photons in a two-dimensional photon fluid," Phys. Rev. A 60, 4114 (1999).
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1998 (4)

N. Akhmediev, W. Krolikowski, and A. W. Snyder, "Partially Coherent Solitons of Variable Shape," Phys. Rev. Lett. 81, 4632 (1998).
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D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, "Theory of Incoherent Dark Solitons," Phys. Rev. Lett. 80, 5113 (1998).
[CrossRef]

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, "Self-trapping of dark incoherent light beams," Science 280, 889 (1998).
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A. W. Snyder and D. J. Mitchell, "Big incoherent solitons," Phys. Rev. Lett. 80, 1422 (1998).
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1997 (4)

M. Mitchell, M. Segev, T. Coskun, D.N. Christodoulides, "Theory of Self-Trapped Spatially Incoherent Light Beams," Phys. Rev. Lett. 79, 4990 (1997).
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D.N. Christodoulides, T.H. Coskun, M. Mitchell, M. Segev, "Theory of Incoherent Self-Focusing in Biased Photorefractive Media," Phys. Rev. Lett. 78, 646 (1997).
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P. A. Robinson, "Nonlinear wave collapse and strong turbulence," Rev. Mod. Phys. 69, 507 (1997).
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M. Mitchell and M. Segev, "Self-trapping of incoherent white light," Nature (London) 387, 880 (1997).
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1996 (1)

M. Mitchell, Z. Chen, Ming-feng Shih, and M. Segev, "Self-Trapping of Partially Spatially Incoherent Light," Phys. Rev. Lett. 77, 490 (1996).
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1995 (1)

S. B. Cavalcanti, G. P. Agrawal, and M. Yu, "Noise amplification in dispersive nonlinear media," Phys. Rev. A 51, 4086 (1995).
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1992 (3)

T. Frisch, Y. Pomeau, and S. Rica "Transition to dissipation in a model of superflow," Phys. Rev. Lett. 69, 1644 (1992).
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S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, "Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schr¨odinger equation," Physica D 57, 96 (1992).
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Y. Pomeau, "Long time behavior of solutions of nonlinear classical field equations: the example of NLS defocusing," Physica D 61, 227 (1992).
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1991 (1)

1988 (1)

V. E. Zakharov, A. N. Pushkarev, V. F. Shvetz, V. V. Yan’kov, "Solitonic turbulence," Pis’ma vZh. Eksp. Teor. Fiz. 48, 79-81 (1988) [JETP Lett. 48, 83-85 (1988)].

1985 (1)

V. E. Zakharov, S. L. Musher and A. M. Rubenchik, "Hamiltonian approach to the description of non-linear plasma phenomena," Phys. Reports 129, 285-366 (1985).
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1977 (1)

A. Hasegawa, "Envelope soliton of random phase waves," Phys. Fluids 20, 2155-2156 (1977).
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1975 (1)

A. Hasegawa, "Dynamics of an ensemble of plane waves in nonlinear dispersive media," Phys. Fluids 18, 77-78 (1975).
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1974 (1)

G.A. Pasmanik, "Self-interaction of incoherent light beams," Sov. Phys. JETP 39, 234 (1974).

1968 (1)

A. C. Newell, "The closure problem in a system of random gravity waves," Rev. of Geophys. 6, 1-31 (1968).
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1964 (1)

J. Ducuing and N . Bloembergen, "Statistical Fluctuations in Nonlinear Optical Processes," Phys. Rev. 133, A1493 - A1502 (1964).
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1963 (1)

K. Hasselmann, "On the non-linear energy transfer in a gravity-wave spectrum. Part 2. Conservation theorems; wave-particle analogy; irreversibility," J. Fluid Mech. 15, 273-281 (1963).
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1962 (1)

K. Hasselmann, "On the non-linear energy transfer in a gravity-wave spectrum. Part 1. General theory," J. Fluid Mech. 12, 481-500 (1962).
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Agrawal, G. P.

S. A. Ponomarenko, N. M. Litchinitser, and G. P. Agrawal "Theory of incoherent optical solitons: Beyond the mean-field approximation," Phys. Rev. E 70, 015603 (2004).
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S. A. Ponomarenko and G. P. Agrawal "Asymmetric incoherent vector solitons," Phys. Rev. E 69, 036604 (2004).
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S. B. Cavalcanti, G. P. Agrawal, and M. Yu, "Noise amplification in dispersive nonlinear media," Phys. Rev. A 51, 4086 (1995).
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Akhmediev, N.

N. Akhmediev, W. Krolikowski, and A. W. Snyder, "Partially Coherent Solitons of Variable Shape," Phys. Rev. Lett. 81, 4632 (1998).
[CrossRef]

Akhmediev, N. N.

A. A. Sukhorukov and N. N. Akhmediev "Coherent and Incoherent Contributions to Multisoliton Complexes," Phys. Rev. Lett. 83, 4736 (1999).
[CrossRef]

Anastassiou, C.

C. Anastassiou, M. Soljacic, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, "Eliminating the Transverse Instabilities of Kerr Solitons," Phys. Rev. Lett. 85, 4888 (2000).
[CrossRef] [PubMed]

Anderson, D.

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, "Features of modulational instability of partially coherent light: Importance of the incoherence spectrum," Phys. Rev. E 69, 025601 (2004).
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L. Helczynski, M. Lisak, and D. Anderson "Influence of higher-order dispersion on modulational instability and pulse broadening of partially incoherent light," Phys. Rev. E 67, 026602 (2003).
[CrossRef]

M. Lisak, L. Helczynski, D. Anderson, "Relation between different formalisms describing partially incoherent wave propagation in nonlinear optical media," Opt. Commun. 220, 321 (2003).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov "Statistical theory for incoherent light propagation in nonlinear media," Phys. Rev. E 65, 035602 (2002).
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Aschieri, P.

A. Picozzi and P. Aschieri, "Influence of dispersion on the resonant interaction between three incoherent waves," Phys. Rev. E 72, 046606 (2005).
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Assanto, G.

Ayanides, J.-P.

Bang, O.

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, "Observation of attraction between Dark Solitons," Phys. Rev. Lett. 96, 043901 (2006).
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W. Krolikowski, D. Edmundson, and O. Bang, "Unified model for partially coherent solitons in logarithmically nonlinear media," Phys. Rev. E 61, 3122 (2000).
[CrossRef]

O. Bang, D. Edmundson, andW. Krolikowski, "Collapse of Incoherent Light Beams in Inertial Bulk Kerr Media," Phys. Rev. Lett. 83, 5479 (1999).
[CrossRef]

Bartal, G.

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, D. N. Christodoulides, "Observation of random-phase lattice solitons," Nature (London) 433, 500 (2005).
[CrossRef] [PubMed]

Berloff, N. G.

N. G. Berloff and B. V. Svistunov "Scenario of strongly nonequilibrated Bose-Einstein condensation," Phys. Rev. A 66, 013603 (2002).
[CrossRef]

Biven, L.

A. C. Newell, S. Nazarenko and L. Biven, "Wave turbulence and intermittency," Physica D 152-153, 520-550 (2001).
[CrossRef]

Bloembergen, N

J. Ducuing and N . Bloembergen, "Statistical Fluctuations in Nonlinear Optical Processes," Phys. Rev. 133, A1493 - A1502 (1964).
[CrossRef]

Boyce, J.

R. Y. Chiao and J. Boyce "Bogoliubov dispersion relation and the possibility of superfluidity for weakly interacting photons in a two-dimensional photon fluid," Phys. Rev. A 60, 4114 (1999).
[CrossRef]

Buljan, H.

O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, "Incoherent solitons in instantaneous nonlocal nonlinear media," Phys. Rev. E 73, 015601 (2006).
[CrossRef]

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, D. N. Christodoulides, "Observation of random-phase lattice solitons," Nature (London) 433, 500 (2005).
[CrossRef] [PubMed]

H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides "Random-Phase Solitons in Nonlinear Periodic Lattices," Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef] [PubMed]

H. Buljan, M. Soljacic, T. Carmon, and M. Segev, "Cavity pattern formation with incoherent light," Phys. Rev. E 68, 016616 (2003).
[CrossRef]

Burnett, K.

M. J. Davis, S. A. Morgan, and K. Burnett, "Simulations of thermal Bose fields in the classical limit," Phys. Rev. A 66, 053618 (2002).
[CrossRef]

M. J. Davis, S. A. Morgan and K. Burnett, "Simulations of Bose Fields at Finite Temperature," Phys. Rev. Lett. 87, 160402 (2001).
[CrossRef] [PubMed]

Carmon, T.

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, D. N. Christodoulides, "Observation of random-phase lattice solitons," Nature (London) 433, 500 (2005).
[CrossRef] [PubMed]

H. Buljan, M. Soljacic, T. Carmon, and M. Segev, "Cavity pattern formation with incoherent light," Phys. Rev. E 68, 016616 (2003).
[CrossRef]

Cavalcanti, S. B.

S. B. Cavalcanti, G. P. Agrawal, and M. Yu, "Noise amplification in dispersive nonlinear media," Phys. Rev. A 51, 4086 (1995).
[CrossRef] [PubMed]

Chen, Z.

T. H. Coskun, D. N. Christodoulides, Z. Chen, and M. Segev "Dark incoherent soliton splitting and phasememory effects: Theory and experiment," Phys. Rev. E 59, R4777 (1999).
[CrossRef]

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, "Self-trapping of dark incoherent light beams," Science 280, 889 (1998).
[CrossRef] [PubMed]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, "Theory of Incoherent Dark Solitons," Phys. Rev. Lett. 80, 5113 (1998).
[CrossRef]

M. Mitchell, Z. Chen, Ming-feng Shih, and M. Segev, "Self-Trapping of Partially Spatially Incoherent Light," Phys. Rev. Lett. 77, 490 (1996).
[CrossRef] [PubMed]

Chiao, R. Y.

R. Y. Chiao, T. H. Hansson, J. M. Leinaas, and S. Viefers, "Effective photon-photon interaction in a twodimensional photon fluid," Phys. Rev. A 69, 063816 (2004).
[CrossRef]

R. Y. Chiao and J. Boyce "Bogoliubov dispersion relation and the possibility of superfluidity for weakly interacting photons in a two-dimensional photon fluid," Phys. Rev. A 60, 4114 (1999).
[CrossRef]

Christodoulides, D. N.

K. G. Makris, H. Sarkissian, D. N. Christodoulides, and G. Assanto "Nonlocal incoherent spatial solitons in liquid crystals," J. Opt. Soc. Am. B 22, 1371-1377 (2005).
[CrossRef]

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, D. N. Christodoulides, "Observation of random-phase lattice solitons," Nature (London) 433, 500 (2005).
[CrossRef] [PubMed]

H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides "Random-Phase Solitons in Nonlinear Periodic Lattices," Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef] [PubMed]

S. M. Sears, M. Soljacic, D. N. Christodoulides, and M. Segev, "Pattern formation via symmetry breaking in nonlinear weakly correlated systems," Phys. Rev. E 65, 036620 (2002).
[CrossRef]

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601 (2001).
[CrossRef]

C. Anastassiou, M. Soljacic, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, "Eliminating the Transverse Instabilities of Kerr Solitons," Phys. Rev. Lett. 85, 4888 (2000).
[CrossRef] [PubMed]

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath "Modulation Instability of Incoherent Beams in Noninstantaneous Nonlinear Media," Phys. Rev. Lett. 84, 467 (2000).
[CrossRef] [PubMed]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams," Science 290, 495-498 (2000).
[CrossRef] [PubMed]

T. H. Coskun, A. G. Grandpierre, D. N. Christodoulides, and M. Segev "Coherence enhancement of spatially incoherent light beams through soliton interactions," Opt. Lett. 25, 826 (2000).
[CrossRef]

T. H. Coskun, D. N. Christodoulides, Z. Chen, and M. Segev "Dark incoherent soliton splitting and phasememory effects: Theory and experiment," Phys. Rev. E 59, R4777 (1999).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, "Theory of Incoherent Dark Solitons," Phys. Rev. Lett. 80, 5113 (1998).
[CrossRef]

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, "Self-trapping of dark incoherent light beams," Science 280, 889 (1998).
[CrossRef] [PubMed]

Christodoulides, D.N.

M. Mitchell, M. Segev, T. Coskun, D.N. Christodoulides, "Theory of Self-Trapped Spatially Incoherent Light Beams," Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

D.N. Christodoulides, T.H. Coskun, M. Mitchell, M. Segev, "Theory of Incoherent Self-Focusing in Biased Photorefractive Media," Phys. Rev. Lett. 78, 646 (1997).
[CrossRef]

Clerici, M.

O. Jedrkiewicz, A. Picozzi, M. Clerici, D. Faccio, and P. Di Trapani, "Emergence of X-Shaped Spatiotemporal Coherence in Optical Waves," Phys. Rev. Lett. 97, 243903 (2006).
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Coen, S.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

Cohen, O.

O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, "Incoherent solitons in instantaneous nonlocal nonlinear media," Phys. Rev. E 73, 015601 (2006).
[CrossRef]

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, D. N. Christodoulides, "Observation of random-phase lattice solitons," Nature (London) 433, 500 (2005).
[CrossRef] [PubMed]

H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides "Random-Phase Solitons in Nonlinear Periodic Lattices," Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef] [PubMed]

Connaughton, C.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, "Condensation of Classical Nonlinear Waves," Phys. Rev. Lett. 95, 263901 (2005).
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Conti, C.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, "Nonlinear Electromagnetic X-Waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated x-shaped light bullets," Phys. Rev. Lett. 91, 093904 (2003);
[CrossRef] [PubMed]

Coskun, T.

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath "Modulation Instability of Incoherent Beams in Noninstantaneous Nonlinear Media," Phys. Rev. Lett. 84, 467 (2000).
[CrossRef] [PubMed]

M. Mitchell, M. Segev, T. Coskun, D.N. Christodoulides, "Theory of Self-Trapped Spatially Incoherent Light Beams," Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

Coskun, T. H.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601 (2001).
[CrossRef]

T. H. Coskun, A. G. Grandpierre, D. N. Christodoulides, and M. Segev "Coherence enhancement of spatially incoherent light beams through soliton interactions," Opt. Lett. 25, 826 (2000).
[CrossRef]

T. H. Coskun, D. N. Christodoulides, Z. Chen, and M. Segev "Dark incoherent soliton splitting and phasememory effects: Theory and experiment," Phys. Rev. E 59, R4777 (1999).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, "Theory of Incoherent Dark Solitons," Phys. Rev. Lett. 80, 5113 (1998).
[CrossRef]

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, "Self-trapping of dark incoherent light beams," Science 280, 889 (1998).
[CrossRef] [PubMed]

Coskun, T.H.

D.N. Christodoulides, T.H. Coskun, M. Mitchell, M. Segev, "Theory of Incoherent Self-Focusing in Biased Photorefractive Media," Phys. Rev. Lett. 78, 646 (1997).
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S. A. Ponomarenko and G. P. Agrawal "Asymmetric incoherent vector solitons," Phys. Rev. E 69, 036604 (2004).
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S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, "Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schr¨odinger equation," Physica D 57, 96 (1992).
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C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, "Condensation of Classical Nonlinear Waves," Phys. Rev. Lett. 95, 263901 (2005).
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H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides "Random-Phase Solitons in Nonlinear Periodic Lattices," Phys. Rev. Lett. 92, 223901 (2004).
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O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, "Incoherent solitons in instantaneous nonlocal nonlinear media," Phys. Rev. E 73, 015601 (2006).
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O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, D. N. Christodoulides, "Observation of random-phase lattice solitons," Nature (London) 433, 500 (2005).
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D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601 (2001).
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C. Anastassiou, M. Soljacic, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, "Eliminating the Transverse Instabilities of Kerr Solitons," Phys. Rev. Lett. 85, 4888 (2000).
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M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath "Modulation Instability of Incoherent Beams in Noninstantaneous Nonlinear Media," Phys. Rev. Lett. 84, 467 (2000).
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T. H. Coskun, D. N. Christodoulides, Z. Chen, and M. Segev "Dark incoherent soliton splitting and phasememory effects: Theory and experiment," Phys. Rev. E 59, R4777 (1999).
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A. Mussot, E. Lantz, H. Maillotte, T. Sylvestre, C. Finot, and S. Pitois, "Spectral broadening of a partially coherent CW laser beam in single-mode optical fibers," Opt. Exp. 12, 2838 (2004).
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C. Anastassiou, M. Soljacic, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, "Eliminating the Transverse Instabilities of Kerr Solitons," Phys. Rev. Lett. 85, 4888 (2000).
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P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated x-shaped light bullets," Phys. Rev. Lett. 91, 093904 (2003);
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C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, "Nonlinear Electromagnetic X-Waves," Phys. Rev. Lett. 90, 170406 (2003).
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M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath "Modulation Instability of Incoherent Beams in Noninstantaneous Nonlinear Media," Phys. Rev. Lett. 84, 467 (2000).
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V. E. Zakharov, A. N. Pushkarev, V. F. Shvetz, V. V. Yan’kov, "Solitonic turbulence," Pis’ma vZh. Eksp. Teor. Fiz. 48, 79-81 (1988) [JETP Lett. 48, 83-85 (1988)].

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T. H. Coskun, D. N. Christodoulides, Z. Chen, and M. Segev "Dark incoherent soliton splitting and phasememory effects: Theory and experiment," Phys. Rev. E 59, R4777 (1999).
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H. Buljan, M. Soljacic, T. Carmon, and M. Segev, "Cavity pattern formation with incoherent light," Phys. Rev. E 68, 016616 (2003).
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M. Mitchell, M. Segev, T. Coskun, D.N. Christodoulides, "Theory of Self-Trapped Spatially Incoherent Light Beams," Phys. Rev. Lett. 79, 4990 (1997).
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A. W. Snyder and D. J. Mitchell, "Big incoherent solitons," Phys. Rev. Lett. 80, 1422 (1998).
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H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides "Random-Phase Solitons in Nonlinear Periodic Lattices," Phys. Rev. Lett. 92, 223901 (2004).
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M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath "Modulation Instability of Incoherent Beams in Noninstantaneous Nonlinear Media," Phys. Rev. Lett. 84, 467 (2000).
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Supplementary Material (2)

» Media 1: AVI (2054 KB)     
» Media 2: AVI (1274 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

Analogy between a system of classical particles (two-particle collision) and the propagation of an optical wave in a cubic (e.g., Kerr) nonlinear medium. As described by the kinetic gas theory, collisions between particles are responsible for an irreversible evolution of the gas towards thermodynamic equilibrium. In complete analogy, an optical wave propagating in a nonlinear medium should exhibit an irreversible evolution to equilibrium, as a result of the four-wave mixing between frequency components of the incoherent field.

Fig. 2.
Fig. 2.

Optical wave condensation: Numerical simulation of the normalized three-dimensional NLS Eq.(1) showing the evolution of the spatial spectrum [S(k,ω = 0,z) =|ψ̃|2 (k,ω= 0,z)] of the field [in normal (a) [Media 1], and logarithmic (b) [Media 2], scales] during its propagation. The concentration of the power of the field in the fundamental mode k =ω= 0 solely results from its natural irreversible evolution towards the equilibrium state. The evolution of some relevant quantities of the field corresponding to this numerical simulation are illustrated in Fig. 3. The complete movie corresponds to a propagation over 500Lnl . (The space-time discretization of the normalized NLS Eq.(1) is dx = dt = 1, the power density is N/V = 1/2, with a computational box of 643).

Fig. 3.
Fig. 3.

Thermal wave relaxation to equilibrium: Evolution of the fraction of condensed power (a), entropy (b), and energy (c,d) corresponding to the numerical simulation illustrated in Fig. 2. The fraction of condensed power irreversibly evolves towards the equilibrium value predicted by the theory (N 0/N ≃ 71% for a total energy of H = 1 and a power density N/V = 1/2). The process of entropy growth is saturated once the equilibrium state is reached, as described by the H-theorem of entropy growth. Figs. (c) and (d) show that a transfer of energy occurs from the nonlinear contribution Hnl to the linear contribution Hl , while the total energy H = Hnl +Hl remains constant. This process of energy transfer explains why an increase of entropy in the field requires the generation of a the coherent structure (i.e., the plane-wave condensate).

Fig. 4.
Fig. 4.

Wave condensation in two-dimensions beyond the thermodynamic limit: Fraction of condensed power N 0/N vs T/Tc calculated by solving the coupled Eqs.(32,33). Even for small values of the area A, the condensation curve exhibits properties analogous to those of the three-dimensional condensation in the thermodynamic limit, as described by Eq.(31).

Fig. 5.
Fig. 5.

Wave condensation in two-dimensions: the continuous line refers to the condensation curve calculated theoretically [Eq.(35)], and points (◇) refer to the numerical simulation of the 2D NLS Eq.(6). Each point (◇) corresponds to an averaging of N 0/N over 1000 length units, once the equilibrium state is reached. (The spatial discretization of the normalized NLS Eq.(6) is dx = 1/2, the power density is N/A = 1/4, the number of points 322).

Equations (37)

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i z ψ = α 2 ψ + β tt ψ + g ψ 2 ψ
N = ψ 2 d r dt ,
H = H l + H nl ,
H l = ( α ψ 2 β t ψ 2 ) d r dt ,
H nl = g 2 ψ 4 d r dt .
i z ψ = α 2 ψ + g ψ 2 ψ .
i z B = α ( 2 2 1 2 ) B + g ( ψ 1 2 ψ 1 * ψ 2 * ψ 2 * 2 ψ 1 ψ 2 ) ,
i z B = 2 α r . y B + 2 g [ B ( z , r 1 , 0 ) B ( z , r 2 , 0 ) ] B ,
i z ψ = α 2 ψ + g ψ 2 ψ ,
n k ( z , r ) = B ( z , r , y ) exp ( i k . y ) d y .
z n k ( z , r ) = 2 α k . r n k ( z , r ) + 2 g p = 0 ( 1 ) p 2 2 p ( 2 p + 1 ) ! 2 p + 1 N ˜ ( z , r ) r 2 p + 1 . 2 p + 1 n k ( z , r ) k 2 p + 1 .
z n k ( z , r ) = 2 α k . r n k + 2 g r N ˜ . k n k ,
N ˜ ( z , r ) = n k ( z , r ) d k .
d z n k ( z , r ) z n + k ˙ . k n + r ˙ . r n = 0 ,
k ˙ = z k = r ω ˜ = r ( 2 g N ˜ ) ,
r ˙ = z r = k ω ˜ = 2 α k ,
ω ˜ ( r , k ) = α k 2 + 2 g N ˜ ( r ) .
i z n k 0 , ω 0 δ k 4 k 0 δ ω 4 ω 0 = g d k 1 3 d ω 1 3 ( J 1,2 3,4 δ 0,2 ; 1,3 J ¯ 1,2 3,0 δ 4,2 ; 1,3 )
z n k 1 , ω 1 ( z ) = 𝒞 oll [ n k 1 , ω 1 ] ,
𝒞 oll [ n k 1 , ω 1 ] = d k 2 4 d ω 2 4 W 0 ( n k 3 , ω 3 n k 4 , ω 4 n k 1 , ω 1 + n k 3 , ω 3 n k 4 , ω 4 n k 2 , ω 2 n k 1 , ω 1 n k 2 , ω 2 n k 3 , ω 3 n k 1 , ω 1 n k 2 , ω 2 n k 4 , ω 4 ) .
W 0 = g 2 δ ( 1 ) ( ω 1 + ω 2 ω 3 ω 4 ) δ ( 2 ) ( k 1 + k 2 k 3 k 4 ) × δ ( 1 ) [ K ( k 1 , ω 1 ) + K ( k 2 , ω 2 ) K ( k 3 , ω 3 ) K ( k 4 , ω 4 ) ] ,
K ( k , ω ) = α k 2 β ω 2 ,
N = V n k , ω ( z ) d k ,
H l = V K ( k , ω ) n k , ω ( z ) d k ,
S ( z ) = V Log [ n k , ω ( z ) ] d k .
n k , ω eq = T K ( k , ω ) μ ,
λ c eq α μ , τ c eq β μ .
N V = 4 πT k c α β [ 1 μ k c arctan ( k c μ ) ] ,
H l V = 4 πT k c 3 3 α β [ 1 + 3 μ k c 2 + 3 ( μ k c 2 ) 3 2 arctan ( k c μ ) ] ,
N 0 N = 1 H l H l , c ,
N 0 N = 1 T T c ,
1 T = 1 k x , k y 1 [ α ( k x 2 + k y 2 ) μ ] ,
N 0 N = 1 T k x , k y 1 [ α ( k x 2 k y 2 ) μ ( T ) ] .
N 0 N 1 T T c ,
H A = ρ 2 + ( ρ ρ 0 ) 2 2 + ( ρ ρ 0 ) k 1 k [ ( α k 2 + ρ 0 ) ω B 2 ( k ) ] ,
T dS = d H l μ dN ,
n k eq = T [ K ( k ) 2 v . k μ ] ,

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