Phase transitions in nuclear reactions
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Phase Transitions in Nuclear Reactions. Definition of phase transitions Non analytical thermo potential Negative heat capacity (1 st order) Liquid gas case C<0 and Volume (order parameter) fluctuations Thermo of nuclear reactions Coulomb reduces coexistence and C<0 region

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Phase Transitions in Nuclear Reactions

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Phase Transitions in Nuclear Reactions

  • Definition of phase transitions

    • Non analytical thermo potential

    • Negative heat capacity (1st order)

  • Liquid gas case

    • C<0 and Volume (order parameter) fluctuations

  • Thermo of nuclear reactions

    • Coulomb reduces coexistence and C<0 region

    • Statistical description of radial flow

  • Conclusion and application to data

  • Ph. Chomaz and F. Gulminelli, Caen France


    I

    -I-

    Definition of phase transition

    • Discontinuity in derivatives of thermo potential when

    • 1st order equivalent to negative heat capacities when


    Phase transition in infinite systems

    Thermodynamical potentials

    non analytical

    L.E. Reichl, Texas Press (1980)

    Caloric curve

    Order of transition:

    discontinuity

    b

    Temperature

    Ehrenfest’s definition

    E1

    E2

    Energy

    Ex: first order:

    EOS discontinuous

    R. Balian, Springer (1982)


    Complex b

    Im(b)

    b

    Re(b)

    • Zeroes of Z reach real axis

    Yang & Lee Phys Rev 87(1952)404

    1st order in finite systems

    PC & Gulminelli Phys A (2003)

    • Order param. free (canonical)


    Complex b

    Im(b)

    b

    Re(b)

    b

    E distribution at

    • Zeroes of Z reach real axis

    Yang & Lee Phys Rev 87(1952)404

    • Bimodal E distribution(P(E))

    Energy

    K.C. Lee Phys Rev E 53 (1996) 6558

    Caloric curve

    b

    Temperature

    E1

    E2

    E1

    E2

    Energy

    • Back Bending in EOS(T(E))

    K. Binder, D.P. Landau Phys Rev B30 (1984) 1477

    Ck(3/2)

    • Abnormal fluctuation (sk(E))

    J.L. Lebowitz (1967), PC & Gulminelli, NPA 647(1999)153

    E1

    E2

    Energy

    1st order in finite systems

    PC & Gulminelli Phys A (2003)

    • Order param. free (canonical)

    • Order param. fixe (microcanonical)

    sk /T2


    II

    -II-

    Liquid gas transition

    • Negative heat capacity in fluctuating volume ensemble

      • Difference between CP and CV

  • Negative heat capacity in statistical models

    • Neg. C and Channel opening


  • Canonical lattice gas

    at constant V

    T=10

    MeV

    8

    Pressure ( MeV/fm3)

    -4 -2 0 2 4 6 8

    6

    4

    0 0.2

    0.4 0.6

    0.8

    r / r

    Density

    0

    Volume: order parameter L-G in a box: V=cst

    Lattice-Gas Model

    • Negative compressibility

    Gulminelli & PC PRL 82(1999)1402


    Canonical lattice gas

    at constant V

    T=10

    MeV

    8

    Pressure ( MeV/fm3)

    Thermo

    limit

    -4 -2 0 2 4 6 8

    6

    4

    Thermo limit

    0 0.2

    0.4 0.6

    0.8

    r / r

    Density

    0

    Volume: order parameter L-G in a box: V=cst

    Lattice-Gas Model

    • CV > 0

    • Negative compressibility

    Gulminelli & PC PRL 82(1999)1402

    See specific discussion for large systems

    Pleimling and Hueller, J.Stat.Phys.104 (2001) 971

    Binder, Physica A 319 (2002) 99.

    Gulminelli et al., cond-mat/0302177, Phys. Rev. E.


    Isobar

    canonical

    lattice

    gas

    Isobar

    ensemble

    P(i) µexp-lV(i)

    Open systems (no box)Fluctuating volume

    PC, Duflot & Gulminelli PRL 85(2000)3587

    • Bimodal P(V)

      • Negative

        compressibility

    • Bimodal P(E)

      • Negative heat

        capacity Cp<0

    microcanonical

    lattice

    gas


    Isochore

    canonical

    lattice

    gas

    Constrain on Vi.e. on order paramameter

    • Suppresses bimodal P(E)

      • No negative heat capacity CV > 0

    microcanonical

    lattice

    gas

    See specific discussion for large systems

    Pleimling and Hueller, J.Stat.Phys.104 (2001) 971

    Binder, Physica A 319 (2002) 99.

    Gulminelli et al., cond-mat/0302177, Phys. Rev. E.


    Probability

    2

    Coulomb interaction VC

    1.6

    1.2

    0.8

    0.4

    Probability

    0

    2

    4

    6

    8

    10

    12

    14

    Nuclear energy

    C<0 in statistical modelsLiquid-gas and channel opening

    • q = 0 =>

      no-Coulomb

  • Many channel opening

  • One Liquid-Gas transition

  • Bimodal P(EN)


    III

    -III-

    Thermo of nuclear reaction

    • Coulomb reduces coexistence and C<0 region

    • Statistical treatment of Coulomb

      • (EN,VC) a common phase diagram for charged and uncharged systems

  • Statistical treatment of radial flow

    • Isobar ensemble required

    • Phase transition not affected


  • Coulomb reduces L-G transition

    • Reduces C<0

      • Lattice-Gas

        • OK up to heavy nuclei

    • Reduces coexistence

      • Bonche-Levit-Vautherin

    Nucl. Phys. A427 (1984) 278

    A=207

    Z= 82

    Gulminelli, PC, Comment to PRC66 (2002) 041601


    Statistical treatment of Non saturating forces interactions

    - Effective charge q = 1 charged system -

    - Effective charge q = 0 uncharged system -

    • A unique framework: e-bE = e-bNEN -bCVC

      • Introduce two temperatures bN=band bC =q2b

      • O two energies ENand VC

    bC = 0 Uncharged

    bC = bNCharged


    2

    1.6

    1.2

    0.8

    0.4

    0

    2

    4

    6

    8

    10

    12

    14

    With and without Coulomb a unique S(EN,VC)

    Probability

    • (EN ,VC) a unique phase diagram

    • Coulomb reduces E bimodality

      • different weight

      • rotation of E axis

    0

    4

    8

    12

    Energy

    Coulomb interaction VC

    bC =  Uncharged

    bC = bNCharged

    E=EN+VC, charged

    E=EN, uncharged

    Energy

    Gulminelli, PC, Raduta, Raduta, submitted to PRL


    0

    0

    2

    2

    4

    4

    6

    6

    8

    8

    10

    10

    12

    12

    14

    14

    Correction of Coulomb effects

    • If events overlap

    Energy distribution: Entropy

    Reconstructed

    Exact

    Probability

    2

    2

    1.6

    1.6

    bC = 0

    Uncharged

    Coulomb interaction VC

    Coulomb interaction VC

    1.2

    1.2

    bC = bN

    Charged

    Reconstruction

    of the uncharged distribution from the charged one

    0.8

    0.8

    Energy

    Energy

    Gulminelli, PC, Raduta, Raduta, submitted to PRL


    4

    bC = 0

    Uncharged

    3

    Coulomb interaction VC

    2

    1

    bC = bNCharged

    0

    2

    4

    6

    8

    10

    12

    14

    Energy

    Partitions may differ for heavy nuclei

    • Channels are different

      (cf fission)

    • Re-weighting impossible

    • However, a unique phase diagram

      (EN VC)

    Gulminelli, PC, Raduta, Raduta, submitted to PRL


    x

    Thermal distribution in the moving frame

    Flow induces

    negative pressure

    z

    Expansion

    Radial flow at “equilibrium”

    • Equilibrium = Max S under constrains

    • Additional constrains: radial flow<pr(r)>

      • Additional Lagrange multiplier(r)

      • Self similar expansion<pr(r)> = m a r =>(r) =  r

    • Requires a confining constrain => isobar ensemble

    Gulminelli, PC, Nucl-Th (2002)


    x

    z

    Expansion

    Radial flow at “equilibrium”

    • Does no affect r partitioning

    • Only reduces the pressure

    • Shifts p distribution

    • Changes fragment distribution: fragmentation less effective

    • Requires a confining constrain => isobar ensemble

    pressure

    Gulminelli, PC, Nucl-Th (2002)


    IV

    -IV-

    Conclusion and application to data

    • Exact theory of phase transition

      • Negative heat capacity in finite systems

  • Liquid-gas and role of volume

    • Negative heat capacity in open systems

  • Role of Coulomb

    • C<0 phenomenology qualitatively preserved

  • Role of flow

    • Thermodynamics of the isobar ensemble


    • Canonical: total energy

    Ztot = Zk Zp

    sE2 = sk2 + sp2

    2b log Zi = Ci /b2 = si2

    • Microcanonical: partial energy

    Wtot = Wk  Wp

    sE2 = 0, sk2 = sp2

    2b log Wtot = -1/CT2 = f(sk2)

    PC, Gulminelli NPA(1999)

    Heat capacity from energy fluctuation


    • Canonical: total energy

    Ztot = Zk Zp

    sE2 = sk2 + sp2

    2b log Zi = Ci /b2 = si2

    • Microcanonical: partial energy

    Wtot = Wk  Wp

    sE2 = 0, sk2 = sp2

    2b log Wtot = -1/CT2 = f(sk2)

    PC, Gulminelli NPA(1999)

    Heat capacity from energy fluctuation


    2/T2

    T

    V=cte

    p=cte

    px

    Tsalis ensemble

    pz

    Transparency

    x

    z

    Expansion

    A robust signal

    • Depends only on the state

    PC, Duflot & Gulminelli PRL 85(2000)3587

    • Out of equilibrium

    • With flow (20%)

    Gulminelli, PC Phisica A (2002)

    Gulminelli, PC Nucl-th 2002


    E

    =

    m

    m

    +

    E

    E

    2

    i

    i

    i

    coul

    coul

    E

    =E

    -E

    *

    1

    2

    <E

    >=<

    a

    > T

    2

    + 3/2 <M-1> T

    1

    i

    i

    Multifragmentation experiment

    Sort events in energy (Calorimetry)

    Reconstruct a freeze-out partition

    1- Primary fragments:

    2- Freeze-out volume:

    => <E1>, 1

    3- Kinetic EOS:

    T, C1


    Ni+Ni, 32 AMeV Quasi-Projectile

    2

    D’Agostino et al.

    Hot Au, ISIS collaboration

    Fluctuation, Heat Capacity

    1

    INDRA

    0

    0

    2

    4

    6

    8

    Excitation Energy (AMeV)

    Bougault et al Xe+Sn central

    ISIS

    ISIS

    INDRA

    Heat capacity from energy fluctuation

    Multics E1=20.3 E2=6.50.7

    Isis E1=2.5 E2=7.

    Indra E2=6.0.5

    MULTICS

    MULTICS

    Excitation Energy (AMeV)


    Au+Au 35 EF=0

    Au+Au 35 EF=1

    Comparisoncentral / peripheral

    peripheral

    M.D ’AgostinoIWM2001

    central

    Ckin=

    dEkin/dT

    Up to  35 A.MeV the flow ambiguity

    is a small effect


    Heat capacity from any fluctuation

    • From kinetic energy


    Heat capacity from any fluctuation

    • From kinetic energy

    • From any correlated observable (Ex. Abig)

    • r = skA /sksA correlation

    • sA “canonical” fluctuations


    Correction of experimental errors

    • Heat capacity from fluctuations

    • scan can be“filtered”

      • (apparatus and procedure)


    Correction of experimental errors

    • Heat capacity from fluctuations

    • scan can be“filtered”

      • (apparatus and procedure)

    Corrected

    freeze-out

    fluctuation

    • sk can be corrected

      • Iterate the procedure

        • Freeze-out reconstruction

        • Decay toward detector

    Reconstructed

    freeze-out

    fluctuation


    No Heat Bath

    E  Etr

    Order parameters: ,

    Heat Bath

     = tr

    = (SA30- SA2-30)/As


    - 1b -Equivalence betweenbimodality of P(b)Zero's of the partition sum Z(bb)


    A alternative definition in the "intensive" ensemble

    • First order phase transition in finite systems

      is defined by

    • A uniform density of zero's of the partition sum on a line crossing the real baxis perpendicularly

    Yang Lee unit circle theorem


    Zero’s of Z()

    P1(E)

    Partition sum

    P2(b)

    Monomodal distribution

    Saddle point approximation

    E1

    K.C.Lee 1996, 2000


    Im(exp(t))

    Re(exp(t))

    Im(b)

    Re(b)

    Zero’s of Z()

    +

    P1(E)

    P2(E)

    Partition sum

    Bimodal distribution

    E1

    E2

    Saddle point approximation

    b

    K.C.Lee 1996, 2000


    Im(b)

    Re(b)

    Zero’s of Z()

    Factorize the N zero’s [-]

    b

    Inverse Laplace

    2 distributions splitted by the latent heat N

    Ph. Ch., F. Gulminelli, 2003


    IV

    -3c-

    Phase transition under flows

    • Example of oriented flow (transparency)

      • Exact thermodynamics

      • Effects on partitions


    px

    pz

    Transparency

    Transparency at “equilibrium”

    {

    t=-1 target

    t =1 projectile

    • Equilibrium = Max S under constrains

    • Additional constrains: memory flow <tpz>

      • Additional Lagrange multiplier

      • EOS leads to <tpz> = Ap0with

    • Affects p space and fragments,

    • Does not affect the thermo if Eflow subtracted

    Thermal distribution in moving frame

    Gulminelli, PC, Nucl-Th (2002)


    px

    pz

    Transparency

    Transparency at “equilibrium”

    • Does no affect r partitioning

    • Shifts p distribution

    • Changes fragment distribution: fragmentation less effective

    • Affects p space and fragments

    Gulminelli, PC, Nucl-Th (2002)


    px

    pz

    Transparency

    Transparency at “equilibrium”

    • Fragmentation less effective

    • Affect observables if Eflow not subtracted

      • Link between <Epot> and <E>

      • T from <Ek>

      • Ex: Fluctuations of Q values

        • All thermal (from E = - 2e)

        • All relative motion

    • Up to 10-15% no effect

    • Affects p space and fragments

    Gulminelli, PC, Nucl-Th (2002)


    IV

    -V-

    Phase transition other signal

    • Bimodalities and conservation laws

      • Diversity of order parameters in finite systems

  • Re-interpretation of fluctuation

    • Possible experimental corrections

    • New measures of heat capacities


  • Order parameters: ,

    Heat Bath

     = tr

    No Heat Bath

    E  Etr

    = (SA30- SA2-30)/As


    - 2b -Pressure dependent EOSCaloric Curves not EOSFluctuations measure EOS


    E T

    V P

    Volume dependent T(E)

    • 2-D EOS

    Temperature

    Pressure

    Energy


    Lattice-Gas Model

    E T

    V P

    Volume dependent T(E)

    • 2-D EOS

    Temperature

    Pressure

    Energy


    Lattice-Gas Model

    E T

    V P

    Volume dependent T(E)

    • 2-D EOS

    Temperature

    Critical point

    First order

    Liquid gas

    Phase Transition

    Pressure

    Energy


    Lattice-Gas Model

    Volume dependent T(E)

    Transformation

    dependent

    Caloric Curve

    - P = cst

    - <V> = cst

    Temperature

    • Negative Heat Capacity

    Pressure

    Energy


    Temperature

    Fluctuations

    Ck

    Volume dependent EOS

    P = cst

    <V> = cst


    Ck

    Volume dependent EOS

    P = cst

    <V> = cst


    P = cst

    <V> = cst

    Volume dependent EOS

    The Caloric curve

    depends on the

    transformation

    C is not dT/dE

    s2measures EOS

    Abnormal in

    Liquid-Gas


    P = cst

    <V> = cst

    Caloric curve are not EOSFluctuations are

    The Caloric curve

    depends on the

    transformation

    C is not dT/dE

    s2measures EOS

    s2abnormal in

    Liquid-Gas


    Isobare

    ensemble

    P(i) µexp-lV(i)

    Liquid-gas

    Phase transition

    and bimodality


    Isobare

    ensemble

    P(i) µexp-lV(i)

    Liquid-gas

    Phase transition

    and bimodality

    Negative heat

    capacity


    Isobare

    ensemble

    P(i) µexp-lV(i)

    Liquid-gas

    Phase transition

    And bimodality

    Negative

    compressibility

    Negative heat

    capacity


    Isobare

    ensemble

    P(i) µexp-lV(i)

    Liquid-gas

    Phase transition

    And bimodality

    Negative

    compressibility

    Negative heat

    capacity

    Order parameter


    Isobare

    ensemble

    P(i) µexp-lV(i)

    Liquid-gas

    Phase transition

    And bimodality

    Negative

    compressibility

    Negative heat

    capacity

    Order parameter


    Isobare

    ensemble

    P(i) µexp-lV(i)

    Multifragmentation

    Liquid

    Gas

    Statistical Model

    (SMM-Raduta)

    b = 3.7 MeV

    A = 50

    Z = 23

    Bimodal

    energy

    distribution

    Channel

    openings


    Equilibrium : What Does It Mean?

    Thermodynamics : one ∞ system

    One ∞ system = ensemble of ∞ sub-systems


    Equilibrium : What Does It Mean?

    Finite system

    Cannot be cut in sub-systems


    Cannot be cut in sub-systems

    Equilibrium : What Does It Mean?

    One small system in time

    if it is ergodic

    Is a statistical ensemble


    Equilibrium : What Does It Mean?

    One small system in time

    if it is ergodic

    if it is chaotic

    Many events


    Statistics from minimum information

    • “Chaos” populates phase space


    Subtract

    pre-

    equili-

    brium

    Correct

    for

    evaporation

    Statistics from minimum information

    • Compatible with freeze-out

    • “Chaos” populates phase space

    BNV


    Statistics from minimum information

    • “Chaos” populates phase space


    Statistics from minimum information

    • Dynamics: global variables

    • Statistics: population

      • Many different ensembles

    • “Chaos” populates phase space

    E

    Microcanonical

    <E>

    Canonical

    V

    Isochore

    <r3>

    Isobare

    <Q2>

    Deformed

    <p.r>

    Expanding

    <A>

    Grand

    <L>

    Rotating

    ...

    Others


    1. Temperature


    • W equiprobable microstates

    Entropy: missing information

    • T is the entropy increase

    L. Boltzmann

    What is temperature ?

    T

    R. Clausius


    What is temperature ?

    • The microcanonical temperature

    S =T-1 E

    S = k logW


    What is temperature ?

    E = Ethermometer + Ebath

    • It is what thermometers measure.

    • The microcanonical temperature

    S =T-1 E

    S = k logW


    What is temperature ?

    E = Ethermometer + Ebath

    • It is what thermometers measure.

    Distribution of microstates

    • The microcanonical temperature

    S =T-1 E

    S = k logW


    P(Ethermo)

    E

    0

    Ethermo

    0

    What is temperature ?

    E = Ethermo + Ebath

    • It is what thermometers measure.

    Distribution of microstates

    P(Ethermo) Wthermo * Wbath

    max P => logWthermo - logWbath=0

    Most probable partition: Tthermo= bath

    • The microcanonical temperature

    S =T-1 E

    S = k logW


    Temperature (Degrees)

    Heat (Calories per grams)

    Phase transition in infinite systems

    Thermodynamical potentials

    non analytical

    Reichl or Diu or …

    Order of transition:

    discontinuity

    Ehrenfest’s definition

    Ex: first order:

    EOS discontinuous

    R. Balian


    Phase transition in infinite systems

    Thermodynamical potentials

    non analytical

    Reichl or Diu or …

    Order of transition =>

    discontinuous derivative

    Ehrenfest’s definition

    Ex: first order =>

    EOS discontinuous

    R. Balian


    Ck T2

    C =

    Ck T2 - s2

    Energy Partition : E = Ek + Ep

    <Ek> Thermometer

    s2Abnormal

    fluctuations

    Negative

    heat capacity

    2


    2/T2

    T

    V=cte

    p=cte

    px

    Tsalis ensemble

    pz

    Transparency

    x

    z

    Expansion

    A robust signal

    • Depends only on the state

    • Out of equilibrium

    • With flow (up to 20%)


    Temperature (Degrees)

    Heat (Calories per grams)

    Phase transition in infinite systems

    Thermodynamical potentials

    non analytical

    Reichl or Diu or …

    Order of transition:

    discontinuity

    Ehrenfest’s definition

    Ex: first order:

    EOS discontinuous

    R. Balian


    E1

    E2

    1st order phase transition in finite systems

    Ph. Ch. & F. Gulminelli Phys A (2003)

    Complex b

    Im(b)

    b

    Zeroes of Z reach on real axis

    Re(b)

    Yang & Lee Phys Rev 87(1952)404

    b

    E distribution at

    Bimodal E distribution

    K.C. Lee Phys Rev E 53 (1996) 6558

    Energy

    Back Bending in EOS(e.g. T(E))

    Caloric curve

    b

    Temperature

    K. Binder, D.P. Landau Phys Rev B30 (1984) 1477

    E1

    E2

    Energy


    E1

    E2

    1st order phase transition in finite systems

    Ph. Ch. & F. Gulminelli Phys A (2003)

    Complex b

    Im(b)

    b

    Zeroes of Z reach on real axis

    Re(b)

    Yang & Lee Phys Rev 87(1952)404

    b

    E distribution at

    Bimodal E distribution

    K.C. Lee Phys Rev E 53 (1996) 6558

    Energy

    Back Bending in EOS(e.g. T(E))

    Negative heat capacity

    Caloric curve

    b

    Temperature

    K. Binder, D.P. Landau Phys Rev B30 (1984) 1477

    E1

    E2

    Energy


    E1

    E2

    1st order in finite systems

    Complex b

    Im(b)

    b

    PC & Gulminelli Phys A (2003)

    Re(b)

    Zeroes of Z reach on real axis

    Yang & Lee Phys Rev 87(1952)404

    b

    E distribution at

    Bimodal E distribution

    K.C. Lee Phys Rev E 53 (1996) 6558

    Energy

    Back Bending in EOS(e.g. T(E))

    Caloric curve

    b

    Temperature

    K. Binder, D.P. Landau Phys Rev B30 (1984) 1477

    E1

    E2

    Energy

    Abnormal partial fluctuation

    sk /T2

    Ck(3/2)

    J.L. Lebowitz (1967), PC & Gulminelli, NPA 647(1999)153

    E1

    E2

    Energy


    Zeroes of Z(b)

    Im(b)

    b

    Re(b)

    E1

    E2

    1st order phase transition in finite systems

    Ph. Ch. & F. Gulminelli Phys A (2003)

    Zeroes of Z in complex b

    plane converge on real axis

    Yang & Lee Phys Rev 87(1952)404

    b

    Bimodal E distribution

    order parameters

    E distribution at

    K.C. Lee Phys Rev E 53 (1996) 6558

    b

    Back Bending in EOS

    Temperature

    E1

    E2

    Energy

    K.C. Lee Phys Rev E 53 (1952) 6558


    Lattice-Gas Model

    Canonical

    • Closest neighbors Interaction e

    • Metropolis Gibbs Equilibrium

    • Negative Susceptibility and Compressibility


    Generalization: inverted curvatures

    Canonical Lattice-Gas

    • Closest neighbors Interaction e

      • Metropolis Gibbs Equilibrium

    • Negative Susceptibility and Compressibility

    Chemical

    Potential

    Canonical Lattice-Gas

    Gulminelli, PC

    PRL99

    Pressure

    Particle Number


    Generalization: inverted curvatures

    Canonical Lattice-Gas

    • Closest neighbors interaction e

      • Metropolis Gibbs equilibrium

    • Negative susceptibility and compressibility

    Chemical

    Potential

    Canonical Lattice-Gas

    Gulminelli, PC

    PRL99

    Pressure

    Particle Number


    Lattice-Gas Model

    T=10

    MeV

    8

    Pressure ( MeV/fm3)

    -4 -2 0 2 4 6 8

    6

    4

    0.4 0.6

    0.8

    0 0.2

    r / r

    Density

    0

    Density:Order parameter:

    • Usually V=cst

      • (“box”)

    • Negative compressibility

    • CV > 0

    Canonical lattice gas

    at constant V


    E1

    E2

    1st order in finite systems

    Complex b

    Im(b)

    b

    PC & Gulminelli Phys A (2003)

    Re(b)

    Zeroes of Z reach real axis

    Yang & Lee Phys Rev 87(1952)404

    Canonical

    order param. free

    b

    E distribution at

    Bimodal E distribution(P(E))

    K.C. Lee Phys Rev E 53 (1996) 6558

    Energy

    Back Bending in EOS(T(E))

    Caloric curve

    b

    Temperature

    Microcanonical

    order param. fixed

    K. Binder, D.P. Landau Phys Rev B30 (1984) 1477

    E1

    E2

    Energy

    Abnormal fluctuation (sk(E))

    sk /T2

    Ck(3/2)

    J.L. Lebowitz (1967), PC & Gulminelli, NPA 647(1999)153

    E1

    E2

    Energy


    Isobar

    ensemble

    P(i) µexp-lV(i)

    Effects of V constraint

    Constraint on energy may suppress the bimodality

    Constraint on the order parameter may suppress the negative heat capacity


    Isobar

    ensemble

    P(i) µexp-lV(i)

    Open systems (no box)Order param. bimodal

    • Bimodal P(V)

      • Negative compressibility

    • Constraint on V (order parameter) may suppress E bimodality

      • CV > 0

    Isobar

    canonical

    lattice gas


    Isobar

    ensemble

    P(i) µexp-lV(i)

    Volume distribution

    Bimodal Event

    Distribution

    Negative

    compressibility

    Negative heat

    capacity

    Order parameter


    E1

    E2

    1st order in finite systems

    Complex b

    Im(b)

    b

    PC & Gulminelli Phys A (2003)

    Re(b)

    • Order param. free (canonical)

    • Zeroes of Z reach real axis

    b

    E distribution at

    Yang & Lee Phys Rev 87(1952)404

    • Bimodal E distribution(P(E))

    Energy

    K.C. Lee Phys Rev E 53 (1996) 6558

    Caloric curve

    b

    Temperature

    • Order param. fixe (microcanonical)

    E1

    E2

    Energy

    • Back Bending in EOS(T(E))

    sk /T2

    K. Binder, D.P. Landau Phys Rev B30 (1984) 1477

    Ck(3/2)

    • Abnormal fluctuation (sk(E))

    J.L. Lebowitz (1967), PC & Gulminelli, NPA 647(1999)153

    E1

    E2

    Energy


    Energy

    should we expect C<0 in multifragmentation?

    r = A/V

    order

    parameter

    with sharp

    boundary

    without

    energy is an order parameter

    C<0

    If V is not

    constrained


    Wk(Ek)Wp(Ep)

    Sk(Ek) +S(Ep)-S(E)

    PE(Ek) = = e

    W(E)

    -1

    -1

    • Most probable Ek : Tk = ∂Esk = ∂ESp = Tp

      • Thermometer

    • Fluctuations s2 :s2 =

      • Heat capacity

      • Gaussian approximation lowest order

    Ck CP

    T-2 (Ck+Cp)

    Energy partition

    • E = Ek + Ep (kinetic+potential)

    Exact for classical gas


    Ck T2

    C =

    Ck T2 - s2

    Energy Partition : E = Ek + Ep

    <Ek> Thermometer

    s2Abnormal

    fluctuations

    Negative

    heat capacity

    2


    2/T2

    T

    V=cte

    p=cte

    px

    Tsalis ensemble

    pz

    Transparency

    x

    z

    Expansion

    A robust signal

    • Depends only on the state

    • Out of equilibrium

    • With flow (up to 20%)


    An observable: partial energy fluctuations

    <Ek> = 3N T/2 classical

    <Ek> =<iai>T2+3<m>T/2

    Fermi

    Canonical

    Ztot = ZkZp

    sE2 = sk2 + sp2

    2b lnZk = Ck /b2 = sk2

    2b lnZtot = C/b2 = sE2

    Microcanonical

    Wtot = Wk Wp

    sE2 = 0

    sk2 = sp2

    2b lnWtot = -(CT2)-1 = f(sk2)


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