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

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


    Phase transitions in nuclear reactions

    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

    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)


    1st order in finite systems

    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)


    1st order in finite systems1

    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


    Phase transitions in nuclear reactions

    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


  • Volume order parameter l g in a box v cst

    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


    Volume order parameter l g in a box v cst1

    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.


    Open systems no box fluctuating volume

    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


    Constrain on v i e on order paramameter

    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.


    C 0 in statistical models liquid gas and channel opening

    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)


    Phase transitions in nuclear reactions

    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

    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

    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


    With and without coulomb a unique s e n v c

    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


    Correction of coulomb effects

    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


    Partitions may differ for heavy nuclei

    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


    Radial flow at equilibrium

    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)


    Radial flow at equilibrium1

    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)


    Phase transitions in nuclear reactions

    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


  • 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


    Heat capacity from energy fluctuation1

    • 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


    A robust signal

    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


    Multifragmentation experiment

    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


    Heat capacity from energy fluctuation2

    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)


    Phase transitions in nuclear reactions

    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

    Heat capacity from any fluctuation

    • From kinetic energy


    Heat capacity from any fluctuation1

    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

    Correction of experimental errors

    • Heat capacity from fluctuations

    • scan can be“filtered”

      • (apparatus and procedure)


    Correction of experimental errors1

    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


    Phase transitions in nuclear reactions

    No Heat Bath

    E  Etr

    Order parameters: ,

    Heat Bath

     = tr

    = (SA30- SA2-30)/As


    1b equivalence between bimodality of p b zero s of the partition sum z b b

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


    A alternative definition in the intensive ensemble

    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

    Zero’s of Z()

    P1(E)

    Partition sum

    P2(b)

    Monomodal distribution

    Saddle point approximation

    E1

    K.C.Lee 1996, 2000


    Zero s of z1

    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


    Zero s of z2

    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


    Phase transitions in nuclear reactions

    IV

    -3c-

    Phase transition under flows

    • Example of oriented flow (transparency)

      • Exact thermodynamics

      • Effects on partitions


    Transparency at equilibrium

    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)


    Transparency at equilibrium1

    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)


    Transparency at equilibrium2

    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)


    Phase transitions in nuclear reactions

    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


  • Phase transitions in nuclear reactions

    Order parameters: ,

    Heat Bath

     = tr

    No Heat Bath

    E  Etr

    = (SA30- SA2-30)/As


    2b pressure dependent eos caloric curves not eos fluctuations measure eos

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


    Volume dependent t e

    E T

    V P

    Volume dependent T(E)

    • 2-D EOS

    Temperature

    Pressure

    Energy


    Volume dependent t e1

    Lattice-Gas Model

    E T

    V P

    Volume dependent T(E)

    • 2-D EOS

    Temperature

    Pressure

    Energy


    Volume dependent t e2

    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


    Volume dependent t e3

    Lattice-Gas Model

    Volume dependent T(E)

    Transformation

    dependent

    Caloric Curve

    - P = cst

    - <V> = cst

    Temperature

    • Negative Heat Capacity

    Pressure

    Energy


    Volume dependent eos

    Temperature

    Fluctuations

    Ck

    Volume dependent EOS

    P = cst

    <V> = cst


    Phase transitions in nuclear reactions

    Ck

    Volume dependent EOS

    P = cst

    <V> = cst


    Phase transitions in nuclear reactions

    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


    Phase transitions in nuclear reactions

    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


    Liquid gas

    Isobare

    ensemble

    P(i) µexp-lV(i)

    Liquid-gas

    Phase transition

    and bimodality


    Liquid gas1

    Isobare

    ensemble

    P(i) µexp-lV(i)

    Liquid-gas

    Phase transition

    and bimodality

    Negative heat

    capacity


    Liquid gas2

    Isobare

    ensemble

    P(i) µexp-lV(i)

    Liquid-gas

    Phase transition

    And bimodality

    Negative

    compressibility

    Negative heat

    capacity


    Liquid gas3

    Isobare

    ensemble

    P(i) µexp-lV(i)

    Liquid-gas

    Phase transition

    And bimodality

    Negative

    compressibility

    Negative heat

    capacity

    Order parameter


    Liquid gas4

    Isobare

    ensemble

    P(i) µexp-lV(i)

    Liquid-gas

    Phase transition

    And bimodality

    Negative

    compressibility

    Negative heat

    capacity

    Order parameter


    Multifragmentation

    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

    Equilibrium : What Does It Mean?

    Thermodynamics : one ∞ system

    One ∞ system = ensemble of ∞ sub-systems


    Equilibrium what does it mean1

    Equilibrium : What Does It Mean?

    Finite system

    Cannot be cut in sub-systems


    Equilibrium what does it mean2

    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 mean3

    Equilibrium : What Does It Mean?

    One small system in time

    if it is ergodic

    if it is chaotic

    Many events


    Statistics from minimum information

    Statistics from minimum information

    • “Chaos” populates phase space


    Statistics from minimum information1

    Subtract

    pre-

    equili-

    brium

    Correct

    for

    evaporation

    Statistics from minimum information

    • Compatible with freeze-out

    • “Chaos” populates phase space

    BNV


    Statistics from minimum information2

    Statistics from minimum information

    • “Chaos” populates phase space


    Statistics from minimum information3

    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

    1. Temperature


    Phase transitions in nuclear reactions

    • W equiprobable microstates

    Entropy: missing information

    • T is the entropy increase

    L. Boltzmann

    What is temperature ?

    T

    R. Clausius


    Phase transitions in nuclear reactions

    What is temperature ?

    • The microcanonical temperature

    S =T-1 E

    S = k logW


    Phase transitions in nuclear reactions

    What is temperature ?

    E = Ethermometer + Ebath

    • It is what thermometers measure.

    • The microcanonical temperature

    S =T-1 E

    S = k logW


    Phase transitions in nuclear reactions

    What is temperature ?

    E = Ethermometer + Ebath

    • It is what thermometers measure.

    Distribution of microstates

    • The microcanonical temperature

    S =T-1 E

    S = k logW


    Phase transitions in nuclear reactions

    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


    Phase transition in infinite systems1

    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 systems2

    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


    Energy partition e e k e p

    Ck T2

    C =

    Ck T2 - s2

    Energy Partition : E = Ek + Ep

    <Ek> Thermometer

    s2Abnormal

    fluctuations

    Negative

    heat capacity

    2


    A robust signal1

    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%)


    Phase transition in infinite systems3

    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


    1st order phase transition in finite systems

    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


    1st order phase transition in finite systems1

    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


    1st order in finite systems2

    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


    1st order phase transition in finite systems2

    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

    Lattice-Gas Model

    Canonical

    • Closest neighbors Interaction e

    • Metropolis Gibbs Equilibrium

    • Negative Susceptibility and Compressibility


    Generalization inverted curvatures

    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 curvatures1

    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


    Density order parameter

    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


    1st order in finite systems3

    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


    Effects of v constraint

    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


    Open systems no box order param bimodal

    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


    Volume distribution

    Isobar

    ensemble

    P(i) µexp-lV(i)

    Volume distribution

    Bimodal Event

    Distribution

    Negative

    compressibility

    Negative heat

    capacity

    Order parameter


    1st order in finite systems4

    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


    Should we expect c 0 in multifragmentation

    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


    Energy partition

    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


    Energy partition e e k e p1

    Ck T2

    C =

    Ck T2 - s2

    Energy Partition : E = Ek + Ep

    <Ek> Thermometer

    s2Abnormal

    fluctuations

    Negative

    heat capacity

    2


    A robust signal2

    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

    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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