KKSMultiACBulkF

KKS model kernel (part 1 of 2) for the Bulk Allen-Cahn. This includes all terms NOT dependent on chemical potential.

Residual

For the 3-phase KKS model, if the non-linear variable is $var element = document.getElementById("moose-equation-8f26e315-c066-4208-a4ce-7777993ba787");katex.render("\\eta_1", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:false});$ ,

$(1)var element = document.getElementById("moose-equation-8181bb3f-95c6-44ec-aa32-bc5eb37948ce");katex.render("R = \\left(\\frac{\\partial h_1}{\\partial \\eta_1} F_1 + \\frac{\\partial h_2}{\\partial \\eta_1} F_2 + \\frac{\\partial h_3}{\\partial \\eta_1} F_3 + W_1 \\frac{\\partial g_1}{\\partial \\eta_1} \\right)", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:true});$

where $var element = document.getElementById("moose-equation-565d5cf7-c385-4f61-9e57-cdc1e214aea4");katex.render("c_i", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:false});$ is the phase concentration for phase $var element = document.getElementById("moose-equation-ca5dba7a-fa39-4583-a722-2b5d36de44eb");katex.render("i", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:false});$ and $var element = document.getElementById("moose-equation-59d4e778-baf0-42d1-afeb-fc5e0347f376");katex.render("h_i", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:false});$ is the interpolation function for phase $var element = document.getElementById("moose-equation-cccbbac4-f085-4667-a983-4f4414378f26");katex.render("i", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:false});$ defined in Folch and Plapp (2005) (referred to as $var element = document.getElementById("moose-equation-145cfa33-796e-4ad0-94d2-695e084e5d46");katex.render("g_i", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:false});$ there, but we use $var element = document.getElementById("moose-equation-3e929154-fd4b-4936-8305-78b6383f49e5");katex.render("h_i", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:false});$ to maintain consistency with other interpolation functions in MOOSE). Here $var element = document.getElementById("moose-equation-c986afe9-6b38-4105-a4fd-a26e4d6a5264");katex.render("g_i = \\eta_i^2 (1-\\eta_i)^2", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:false});$ , also for consistency with notation in MOOSE. $var element = document.getElementById("moose-equation-b04995e8-b3dd-4bae-90b8-82bb9c229aed");katex.render("W_1", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:false});$ is the free energy barrier height.

Jacobian

On-diagonal

If the non-linear variable is $var element = document.getElementById("moose-equation-da676305-983b-4450-9acb-884022ded334");katex.render("\\eta_1", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:false});$ , the on-diagonal Jacobian is

$(2)var element = document.getElementById("moose-equation-42fb024b-04c7-4024-96cb-1e3fb7e9f24d");katex.render("\\begin{aligned} J &=& \\phi_j \\frac{\\partial R}{\\partial \\eta_1} \\\\ &=& \\phi_j \\left( \\frac{\\partial ^2 h_1}{\\partial \\eta_1^2} F_1 + \\frac{\\partial ^2 h_2}{\\partial \\eta_1^2} F_2 + \\frac{\\partial ^2 h_3}{\\partial \\eta_1^2} F_3 + W_1 \\frac{\\partial ^2 g}{\\partial \\eta_1^2} \\right) \\end{aligned}", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:true});$

Off-diagonal

Off-diagonal Jacobian for $var element = document.getElementById("moose-equation-f50176f6-fa68-472b-aa5a-d02bf3ce8b2a");katex.render("\\eta_2", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:false});$ (similar for $var element = document.getElementById("moose-equation-6596b7ab-d792-4f30-b998-3c5adbe5c355");katex.render("\\eta_3", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:false});$ ):

$(3)var element = document.getElementById("moose-equation-b81f29ae-f074-4c72-8f9f-e48c84e2ca8d");katex.render("\\begin{aligned} J &=& \\phi_j \\frac{\\partial R}{\\partial \\eta_2} \\\\ &=& \\phi_j \\left( \\frac{\\partial ^2 h_1}{\\partial \\eta_1 \\partial \\eta_2} F_1 + \\frac{\\partial ^2 h_2}{\\partial \\eta_1 \\partial \\eta_2} F_2 + \\frac{\\partial ^2 h_3}{\\partial \\eta_1 \\partial \\eta_2} F_3 \\right) \\end{aligned}", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:true});$

Off-diagonal Jacobian for $var element = document.getElementById("moose-equation-7ba9dd69-c902-439e-9734-a229544a0002");katex.render("c_1", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:false});$ (similar for $var element = document.getElementById("moose-equation-4e7972d7-e543-454c-8118-4a872c0f7d69");katex.render("c_2, c_3", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:false});$ ):

$(4)var element = document.getElementById("moose-equation-1c13bdfb-2321-446b-b355-1a600c7215b5");katex.render("\\begin{aligned} J &=& \\phi_j \\frac{\\partial R}{\\partial c_1} \\\\ &=& \\phi_j \\frac{\\partial h_1}{\\partial \\eta_1} \\frac{\\partial F_1}{\\partial c_1} \\end{aligned}", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:true});$

These statements can be generalized for non-linear variable $var element = document.getElementById("moose-equation-527bbedb-f190-44f5-b42c-b12ef8d1e0f3");katex.render("v", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:false});$ as:

$(5)var element = document.getElementById("moose-equation-5dce6b94-b528-430b-a950-21386ade081e");katex.render("\\begin{aligned} J &=& \\phi_j \\frac{\\partial R}{\\partial v} \\\\ &=& \\left( \\frac{\\partial ^2 h_1}{\\partial \\eta_1 \\partial v} F_1 + \\frac{\\partial ^2 h_2}{\\partial \\eta_1 \\partial v} F_2 + \\frac{\\partial ^2 h_3}{\\partial \\eta_1 \\partial v} F_3 + \\frac{\\partial h_1}{\\partial \\eta_1} \\frac{\\partial F_1}{\\partial v} + \\frac{\\partial h_2}{\\partial \\eta_1} \\frac{\\partial F_2}{\\partial v} + \\frac{\\partial h_3}{\\partial \\eta_1} \\frac{\\partial F_3}{\\partial v}\\right) \\end{aligned}", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:true});$

For the off-diagonal Jacobians we also need to multiply by $var element = document.getElementById("moose-equation-e80a6e3f-1648-411d-9598-38077ac9ad20");katex.render("L", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:false});$ , the Allen-Cahn mobility.

Input Parameters

eta_iOrder parameter that derivatives are taken with respect to
C++ Type:std::vector
Options:
Description:Order parameter that derivatives are taken with respect to
wiDouble well height parameter
C++ Type:double
Options:
Description:Double well height parameter
variableThe name of the variable that this Kernel operates on
C++ Type:NonlinearVariableName
Options:
Description:The name of the variable that this Kernel operates on
gi_nameBase name for the double well function g_i(eta_i)
C++ Type:MaterialPropertyName
Options:
Description:Base name for the double well function g_i(eta_i)
hj_namesSwitching Function Materials that provide h. Place in same order as Fj_names!
C++ Type:std::vector
Options:
Description:Switching Function Materials that provide h. Place in same order as Fj_names!
Fj_namesList of free energies for each phase. Place in same order as hj_names!
C++ Type:std::vector
Options:
Description:List of free energies for each phase. Place in same order as hj_names!

Required Parameters

mob_nameLThe mobility used with the kernel
Default:L
C++ Type:MaterialPropertyName
Options:
Description:The mobility used with the kernel
argsVector of arguments of the mobility
C++ Type:std::vector
Options:
Description:Vector of arguments of the mobility
displacementsThe displacements
C++ Type:std::vector
Options:
Description:The displacements
blockThe list of block ids (SubdomainID) that this object will be applied
C++ Type:std::vector
Options:
Description:The list of block ids (SubdomainID) that this object will be applied

Optional Parameters

enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Options:
Description:Set the enabled status of the MooseObject.
save_inThe name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector
Options:
Description:The name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False
C++ Type:bool
Options:
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector
Options:
Description:Adds user-defined labels for accessing object parameters via control logic.
seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Options:
Description:The seed for the master random number generator
diag_save_inThe name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector
Options:
Description:The name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Options:
Description:Determines whether this object is calculated using an implicit or explicit form

Advanced Parameters

vector_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime
C++ Type:MultiMooseEnum
Options:nontime time
Description:The tag for the vectors this Kernel should fill
extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector
Options:
Description:The extra tags for the vectors this Kernel should fill
matrix_tagssystemThe tag for the matrices this Kernel should fill
Default:system
C++ Type:MultiMooseEnum
Options:nontime system
Description:The tag for the matrices this Kernel should fill
extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector
Options:
Description:The extra tags for the matrices this Kernel should fill

Tagging Parameters

Input Files

modules/phase_field/test/tests/KKS_system/kks_multiphase.i

modules/phase_field/test/tests/KKS_system/kks_multiphase.i

#
# This test is for the 3-phase KKS model
#

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 20
  ny = 20
  nz = 0
  xmin = 0
  xmax = 40
  ymin = 0
  ymax = 40
  zmin = 0
  zmax = 0
  elem_type = QUAD4
[]

[BCs]
  [./Periodic]
    [./all]
      auto_direction = 'x y'
    [../]
  [../]
[]

[AuxVariables]
  [./Energy]
    order = CONSTANT
    family = MONOMIAL
  [../]
[]

[Variables]
  # concentration
  [./c]
    order = FIRST
    family = LAGRANGE
  [../]

  # order parameter 1
  [./eta1]
    order = FIRST
    family = LAGRANGE
  [../]

  # order parameter 2
  [./eta2]
    order = FIRST
    family = LAGRANGE
  [../]

  # order parameter 3
  [./eta3]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.0
  [../]

  # phase concentration 1
  [./c1]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.2
  [../]

  # phase concentration 2
  [./c2]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.5
  [../]

  # phase concentration 3
  [./c3]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.8
  [../]

  # Lagrange multiplier
  [./lambda]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.0
  [../]
[]

[ICs]
  [./eta1]
    variable = eta1
    type = SmoothCircleIC
    x1 = 20.0
    y1 = 20.0
    radius = 10
    invalue = 0.9
    outvalue = 0.1
    int_width = 4
  [../]
  [./eta2]
    variable = eta2
    type = SmoothCircleIC
    x1 = 20.0
    y1 = 20.0
    radius = 10
    invalue = 0.1
    outvalue = 0.9
    int_width = 4
  [../]
  [./c]
    variable = c
    type = SmoothCircleIC
    x1 = 20.0
    y1 = 20.0
    radius = 10
    invalue = 0.2
    outvalue = 0.5
    int_width = 2
  [../]
[]


[Materials]
  # simple toy free energies
  [./f1]
    type = DerivativeParsedMaterial
    f_name = F1
    args = 'c1'
    function = '20*(c1-0.2)^2'
  [../]
  [./f2]
    type = DerivativeParsedMaterial
    f_name = F2
    args = 'c2'
    function = '20*(c2-0.5)^2'
  [../]
  [./f3]
    type = DerivativeParsedMaterial
    f_name = F3
    args = 'c3'
    function = '20*(c3-0.8)^2'
  [../]

  # Switching functions for each phase
  # h1(eta1, eta2, eta3)
  [./h1]
    type = SwitchingFunction3PhaseMaterial
    eta_i = eta1
    eta_j = eta2
    eta_k = eta3
    f_name = h1
  [../]
  # h2(eta1, eta2, eta3)
  [./h2]
    type = SwitchingFunction3PhaseMaterial
    eta_i = eta2
    eta_j = eta3
    eta_k = eta1
    f_name = h2
  [../]
  # h3(eta1, eta2, eta3)
  [./h3]
    type = SwitchingFunction3PhaseMaterial
    eta_i = eta3
    eta_j = eta1
    eta_k = eta2
    f_name = h3
  [../]

  # Coefficients for diffusion equation
  [./Dh1]
    type = DerivativeParsedMaterial
    material_property_names = 'D h1'
    function = D*h1
    f_name = Dh1
  [../]
  [./Dh2]
    type = DerivativeParsedMaterial
    material_property_names = 'D h2'
    function = D*h2
    f_name = Dh2
  [../]
  [./Dh3]
    type = DerivativeParsedMaterial
    material_property_names = 'D h3'
    function = D*h3
    f_name = Dh3
  [../]

  # Barrier functions for each phase
  [./g1]
    type = BarrierFunctionMaterial
    g_order = SIMPLE
    eta = eta1
    function_name = g1
  [../]
  [./g2]
    type = BarrierFunctionMaterial
    g_order = SIMPLE
    eta = eta2
    function_name = g2
  [../]
  [./g3]
    type = BarrierFunctionMaterial
    g_order = SIMPLE
    eta = eta3
    function_name = g3
  [../]

  # constant properties
  [./constants]
    type = GenericConstantMaterial
    prop_names  = 'L   kappa  D'
    prop_values = '0.7 1.0    1'
  [../]
[]

[Kernels]
  #Kernels for diffusion equation
  [./diff_time]
    type = TimeDerivative
    variable = c
  [../]
  [./diff_c1]
    type = MatDiffusion
    variable = c
    D_name = Dh1
    conc = c1
  [../]
  [./diff_c2]
    type = MatDiffusion
    variable = c
    D_name = Dh2
    conc = c2
  [../]
  [./diff_c3]
    type = MatDiffusion
    variable = c
    D_name = Dh3
    conc = c3
  [../]

  # Kernels for Allen-Cahn equation for eta1
  [./deta1dt]
    type = TimeDerivative
    variable = eta1
  [../]
  [./ACBulkF1]
    type = KKSMultiACBulkF
    variable  = eta1
    Fj_names  = 'F1 F2 F3'
    hj_names  = 'h1 h2 h3'
    gi_name   = g1
    eta_i     = eta1
    wi        = 1.0
    args      = 'c1 c2 c3 eta2 eta3'
  [../]
  [./ACBulkC1]
    type = KKSMultiACBulkC
    variable  = eta1
    Fj_names  = 'F1 F2 F3'
    hj_names  = 'h1 h2 h3'
    cj_names  = 'c1 c2 c3'
    eta_i     = eta1
    args      = 'eta2 eta3'
  [../]
  [./ACInterface1]
    type = ACInterface
    variable = eta1
    kappa_name = kappa
  [../]
  [./multipler1]
    type = MatReaction
    variable = eta1
    v = lambda
    mob_name = L
  [../]

  # Kernels for Allen-Cahn equation for eta2
  [./deta2dt]
    type = TimeDerivative
    variable = eta2
  [../]
  [./ACBulkF2]
    type = KKSMultiACBulkF
    variable  = eta2
    Fj_names  = 'F1 F2 F3'
    hj_names  = 'h1 h2 h3'
    gi_name   = g2
    eta_i     = eta2
    wi        = 1.0
    args      = 'c1 c2 c3 eta1 eta3'
  [../]
  [./ACBulkC2]
    type = KKSMultiACBulkC
    variable  = eta2
    Fj_names  = 'F1 F2 F3'
    hj_names  = 'h1 h2 h3'
    cj_names  = 'c1 c2 c3'
    eta_i     = eta2
    args      = 'eta1 eta3'
  [../]
  [./ACInterface2]
    type = ACInterface
    variable = eta2
    kappa_name = kappa
  [../]
  [./multipler2]
    type = MatReaction
    variable = eta2
    v = lambda
    mob_name = L
  [../]

  # Kernels for the Lagrange multiplier equation
  [./mult_lambda]
    type = MatReaction
    variable = lambda
    mob_name = 3
  [../]
  [./mult_ACBulkF_1]
    type = KKSMultiACBulkF
    variable  = lambda
    Fj_names  = 'F1 F2 F3'
    hj_names  = 'h1 h2 h3'
    gi_name   = g1
    eta_i     = eta1
    wi        = 1.0
    mob_name  = 1
    args      = 'c1 c2 c3 eta2 eta3'
  [../]
  [./mult_ACBulkC_1]
    type = KKSMultiACBulkC
    variable  = lambda
    Fj_names  = 'F1 F2 F3'
    hj_names  = 'h1 h2 h3'
    cj_names  = 'c1 c2 c3'
    eta_i     = eta1
    args      = 'eta2 eta3'
    mob_name  = 1
  [../]
  [./mult_CoupledACint_1]
    type = SimpleCoupledACInterface
    variable = lambda
    v = eta1
    kappa_name = kappa
    mob_name = 1
  [../]
  [./mult_ACBulkF_2]
    type = KKSMultiACBulkF
    variable  = lambda
    Fj_names  = 'F1 F2 F3'
    hj_names  = 'h1 h2 h3'
    gi_name   = g2
    eta_i     = eta2
    wi        = 1.0
    mob_name  = 1
    args      = 'c1 c2 c3 eta1 eta3'
  [../]
  [./mult_ACBulkC_2]
    type = KKSMultiACBulkC
    variable  = lambda
    Fj_names  = 'F1 F2 F3'
    hj_names  = 'h1 h2 h3'
    cj_names  = 'c1 c2 c3'
    eta_i     = eta2
    args      = 'eta1 eta3'
    mob_name  = 1
  [../]
  [./mult_CoupledACint_2]
    type = SimpleCoupledACInterface
    variable = lambda
    v = eta2
    kappa_name = kappa
    mob_name = 1
  [../]
  [./mult_ACBulkF_3]
    type = KKSMultiACBulkF
    variable  = lambda
    Fj_names  = 'F1 F2 F3'
    hj_names  = 'h1 h2 h3'
    gi_name   = g3
    eta_i     = eta3
    wi        = 1.0
    mob_name  = 1
    args      = 'c1 c2 c3 eta1 eta2'
  [../]
  [./mult_ACBulkC_3]
    type = KKSMultiACBulkC
    variable  = lambda
    Fj_names  = 'F1 F2 F3'
    hj_names  = 'h1 h2 h3'
    cj_names  = 'c1 c2 c3'
    eta_i     = eta3
    args      = 'eta1 eta2'
    mob_name  = 1
  [../]
  [./mult_CoupledACint_3]
    type = SimpleCoupledACInterface
    variable = lambda
    v = eta3
    kappa_name = kappa
    mob_name = 1
  [../]

  # Kernels for constraint equation eta1 + eta2 + eta3 = 1
  # eta3 is the nonlinear variable for the constraint equation
  [./eta3reaction]
    type = MatReaction
    variable = eta3
    mob_name = 1
  [../]
  [./eta1reaction]
    type = MatReaction
    variable = eta3
    v = eta1
    mob_name = 1
  [../]
  [./eta2reaction]
    type = MatReaction
    variable = eta3
    v = eta2
    mob_name = 1
  [../]
  [./one]
    type = BodyForce
    variable = eta3
    value = -1.0
  [../]

  # Phase concentration constraints
  [./chempot12]
    type = KKSPhaseChemicalPotential
    variable = c1
    cb       = c2
    fa_name  = F1
    fb_name  = F2
  [../]
  [./chempot23]
    type = KKSPhaseChemicalPotential
    variable = c2
    cb       = c3
    fa_name  = F2
    fb_name  = F3
  [../]
  [./phaseconcentration]
    type = KKSMultiPhaseConcentration
    variable = c3
    cj = 'c1 c2 c3'
    hj_names = 'h1 h2 h3'
    etas = 'eta1 eta2 eta3'
    c = c
  [../]
[]

[AuxKernels]
  [./Energy_total]
    type = KKSMultiFreeEnergy
    Fj_names = 'F1 F2 F3'
    hj_names = 'h1 h2 h3'
    gj_names = 'g1 g2 g3'
    variable = Energy
    w = 1
    interfacial_vars =  'eta1  eta2  eta3'
    kappa_names =       'kappa kappa kappa'
  [../]
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'
  petsc_options_iname = '-pc_type -sub_pc_type   -sub_pc_factor_shift_type'
  petsc_options_value = 'asm       ilu            nonzero'
  l_max_its = 30
  nl_max_its = 10
  l_tol = 1.0e-4
  nl_rel_tol = 1.0e-10
  nl_abs_tol = 1.0e-11

  num_steps = 2
  dt = 0.5
[]

[Preconditioning]
  active = 'full'
  [./full]
    type = SMP
    full = true
  [../]
  [./mydebug]
    type = FDP
    full = true
  [../]
[]

[Outputs]
  exodus = true
[]

R. Folch and M. Plapp. Quantitative phase-field modeling of two-phase growth. Phys. Rev. E, 72:011602, Jul 2005. URL: https://link.aps.org/doi/10.1103/PhysRevE.72.011602, doi:10.1103/PhysRevE.72.011602.

@article{Folch05,
    author = "Folch, R. and Plapp, M.",
    title = "Quantitative phase-field modeling of two-phase growth",
    journal = "Phys. Rev. E",
    volume = "72",
    issue = "1",
    pages = "011602",
    numpages = "27",
    year = "2005",
    month = "Jul",
    publisher = "American Physical Society",
    doi = "10.1103/PhysRevE.72.011602",
    url = "https://link.aps.org/doi/10.1103/PhysRevE.72.011602"
}

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