Generalized Langevin Equation and non-Markovian fluctuation-dissipation theorem for particle-bath systems in external oscillating fields
Bingyu Cui Alessio Zaccone







plain

    k̨r̊⟨⟩*theorem*TheoremtheoremTheorem[section]
    
    ^1Statistical Physics Group, Department of Chemical
    Engineering and Biotechnology, University of Cambridge, New Museums Site, CB2
    3RA Cambridge, U.K.^2Cavendish Laboratory,University of Cambridge, JJ Thomson
    Avenue, CB30HE Cambridge,
    U.K.
    The Generalized Langevin Equation (GLE) can be derived from a particle-bath
    Hamiltonian, in both classical and quantum dynamics, and provides a route to
    the (both Markovian and non-Markovian) fluctuation-dissipation theorem (FDT).
    All previous studies have focused either on particle-bath systems with
    time-independent external forces only, or on the simplified case where only the
    tagged particle is subject to the external time-dependent oscillatory field.
    Here we extend the GLE and the corresponding FDT for the more general case
    where both the tagged particle and the bath oscillators respond to an external
    oscillatory field. This is the example of a charged or polarisable particle
    immersed in a bath of other particles that are also charged or polarizable,
    under an external AC electric field.
    For this Hamiltonian, we find that the ensemble average of the stochastic force
    is not zero, but proportional to the AC field. The associated FDT reads as
    ⟨ F_P(t)F_P(t')⟩=mk_BTν(t-t')+(γ e)^2E(t)E(t'), where
    F_p is the random force, ν(t-t') is the friction memory function, and
    γ is a numerical prefactor.
    Generalized Langevin Equation and non-Markovian fluctuation-dissipation
    theorem for particle-bath systems in external oscillating fields
        Bingyu Cui^1, Alessio Zaccone^1,2
        
    ========================================================================================================================================
    
    
    
    
    
    
    
    
    
    
    
    The fluctuation-dissipation theorem (FDT) was
    originally formulated in different contexts by Einstein and by Nyquist, and
    generalized by Onsager <cit.> and by Callen and Welton <cit.>.
    Later, the theorem has been further elaborated in many different
    contexts <cit.>.
    FDT stipulates that the response of a system in thermodynamic equilibrium to a
    small applied force is the same as its response to a spontaneous fluctuation,
    thus connecting the linear-response relaxation of a system to equilibrium, with its statistical fluctuation properties in
    equilibrium. FDT applies to both classical and quantum mechanical
    systems <cit.> and has been generalized to non-Markovian processes
    for classical systems by Zwanzig <cit.>. In the latter case, the noise
    is no longer decorrelated in time, and the time correlation of the stochastic
    force is dictated by the time correlation of the friction which plays the role
    of the memory function in the Generalized Langevin Equation (GLE). The
    non-Markovianity arises from the dynamical coupling of the tagged Brownian
    particle with many particles (harmonic oscillators) forming the heat bath. This
    coupling, which in physical systems may be provided by long-range molecular
    interactions, is thus responsible for both the thermal agitation and the
    damping experienced by the tagged particle.
    
    
    
    
    All versions of the GLE, and of the associated FDT which can be derived from
    it that have been considered in the past, are limited to either systems in the
    absence of external time-dependent forces or, if an external time-dependent
    force is present, its action is restricted to the tagged particle,
    leaving the bath oscillators unaffected by the time-dependent external field <cit.>.
    
    This limitation is obviously artificial and not realistic, because in many
    systems there is no reason to justify why the tagged particle is subjected to
    the external time-dependent oscillating (AC) field whereas the bath oscillators remain unaffected by the
    field.
    This limitation clearly leaves out a number of important physical problems,
    where not only the tagged particle is subjected to the AC field, but also the
    particles that constitute the heat bath are subjected to it.
    This situation is clearly encountered in dielectric matter under a uniform AC
    electric field E(t). In this case, every building block (atom, molecule, ion)
    is polarisable or charged such that it is subjected to a force from the
    electric field. Hence, if the bath is constituted by polarisable or charged
    particles, these will also respond to the AC field and it is unphysical to
    neglect the effect of the AC field on the dynamics of the bath oscillators.
    This situation is schematically depicted in Fig. 1.
    
    Below we provide a solution to this problem by formulating a
    Caldeira-Leggett particle-bath Hamiltonian where both the particle and the bath
    oscillators are subjected to the external AC electric field, which is
    explicitly accounted for in both the Hamiltonian of the particle and the
    Hamiltonian of the bath. The two Hamiltonians are then connected via a
    bi-linear coupling as is standard in this type of models.
    We analytically solve the coupled Hamiltonian for the tagged particle to find a
    new GLE, which, for the first time, accounts for the
    effect of the polarization of the bath under the AC field on the dynamics of
    the tagged particle. This is a more general GLE than any of those proposed so
    far, and only in certain limits it reduces to the known forms of the GLE with
    external time-dependent field.
    
    We also derive the associated FDT and find a surprising result: the
    time-correlation of the stochastic force is not just equal to the memory
    function for the friction, but there is an additional term which is
    proportional to the time-correlation of the AC field, and hence to the
    amplitude of the AC field squared. This term is absent in all previously
    studied versions of the FDT. This result shows that the strength of the
    fluctuating force can be controlled by the external force field in the limit of
    sufficiently strong external driving.
    This framework opens new avenues to understanding the macroscopic response of
    complex liquids, plasmas, colloids and other condensed matter under oscillatory
    fields.
    
    
    
    
    
    
    
    
    We study the classical version of the Caldeira-Leggett coupling between the
    tagged particle and a bath of harmonic oscillators, which was actually proposed
    by Zwanzig <cit.>, and add a new term, H_E, which
    contains the force due to the applied AC electric field acting on both the
    tagged particle and the harmonic oscillators:
    
    
        H=H_S+H_B+H_E
    
    where H_S=P^2/2m+V(Q) is the Hamiltonian of the tagged particle without
    external field. The second term is the standard Hamiltonian of the bath of
    harmonic oscillators that are coupled to the tagged particle,
    H_B=1/2∑_α=1^N[P_α^2/m_α+m_αω_α^2(X_α
    -F_α(Q)/ω_α^2)^2], consisting of the
    standard harmonic oscillator expression for each bath oscillator α, and
    of the coupling term between the tagged particle and the α-th bath
    oscillator, which contains the coupling function F_α(Q).
    
    The new term
    
    
        H_E   =-zeΦ(Q,t)-∑_α e_αΦ(X_α,t)
           =-E_0sin(ω t)(zeQ + ∑_α e_α X_α)
    
    represents the influence of electric field on both the tagged particle (first
    term on the RHS) and the bath oscillators (second term on the RHS). Here, e
    is the unit charge and hence ze is the total charge of tagged particle, where
    e.g. z=± 1 for monovalent ions/electrons in a plasma or z=-1 for the case
    of molecular dielectrics where a molecule polarizes into a negatively charged
    electron cloud which oscillates about a positively charged molecular ion;
    e_α is the net charge of bath particle α which is subjected to
    the same polarization. Here, we only consider motion along the direction of
    electric field.
    
    In the Caldeira-Leggett Hamiltonian, the coupling function is taken to be
    linear in the particle coordinate Q, F_α(Q)=c_αQ, where
    c_α is known as the strength of coupling between the tagged atom and
    the α-th bath oscillator.
    
    
    This Hamiltonian leads in a straightforward manner to the following system of
    differential equations:
    
        dQ/dt   =P/m;  dP/dt=-V'(Q)+∑_αm_α
        c_α(X_α-c_α Q/ω_α^2)+zeE_0sin(ω t)
        dX_α/dt   =P_α/m_α;
        dP_α/dt=-m_αω_α^2X_α+m_α c_α
        Q+e_α E_0sin(ω t)
    
    
    From the second line, upon solving the second-order inhomogeneous ODE with the
    Wronskian method, we get
    
        X_α(t)   =X_α(0)cos(ω_α
        t)+P_α(0)sin(ω_α t)/m_αω_α
           +∫_0^t[c_αQ(s)+e_α/m_α E_0sin(ω
        s)]sin(ω_α(t-s))/ω_αds
    
    
    
    
    The integral ∫_0^tc_α
    Q(s)sin(ω_α(t-s))/ω_αds can be evaluated via
    integration by parts. Upon further denoting E_α(t)=∫_0^te_α
    E_0sin(ω s)sin(ω_α(t-s))/ω_αds, we
    obtain
    
        X_α(t)-c_α
        Q(t)/ω_α^2=(X_α(0)-c_α
        Q(0)/ω_α^2)cos(ω_α t)
           +P_α(0)sin(ω_α
        t)/m_αω_α-∫_0^tc_α
        P(s)cos(ω_α(t-s))/mω_α^2ds+E_α(t)/m_α
    
    
    
    Substituting Eq. (5) into the equation for P(t) in Eq. (3), we derive the
    following Generalized Langevin Equation with AC electric field acting on both
    particle and bath oscillators:
    
        dP/dt   =-V'(Q(t))-∑_α∫_0^tm_αcos(ω_α(t-s))/mω_α^2c_α^2P(s)ds+zeE_0sin(ω
        t)
           +∑_α{m_α c_α[X_α(0)-c_α
        Q(0)/ω_α^2]cos(ω_α t)+c_α
        P_α(0)sin(ω_α t)/ω_α+c_α
        E_α(t)}
           =-V'(Q(t))-∫_0^tν(s)m_α P(t-s)/mds+zeE_0sin(ω
        t)+F_P(t).
    
    
    In analogy with Zwanzig <cit.>, we have introduced the noise or stochastic force F_P(t) defined as
    
    
        F_p(t)   =∑_α{m_α c_α[X_α(0)-c_α
        Q(0)/ω_α^2]cos(ω_α t)
           +c_α P_α(0)sin(ω_α t)/ω_α+c_α
        E_α(t)}.
    
    
    One should note that this expression for the stochastic force is identical with
    the one derived by Zwanzig for a particle-bath system without any external AC field,
    except for the important term c_α E_α(t), which is new and contains
    the effect of the external AC field on the bath oscillators dynamics. This is a
    crucial point because the dynamical response of the bath oscillators to the
    external AC field in turn produces a modification of spectral properties of the
    bath fluctuations, and thus leads to a new form of the stochastic force which
    is different from those studied in previous works.
    
    
    
    
    The memory function for the friction ν(t)=∑_αm_α
    c_α^2/mω_α^2cos(ω_α t) is identical to the
    memory function of systems without external time-dependent forces such as the
    one derived by Zwanzig <cit.>.
    
    
    
    The integral in the function E_α(t)=∫_0^te_α E_0sin(ω
    s)sin(ω_α(t-s))/ω_αds, can be evaluated using
    trigonometric identities which leads to
    
        E_α(t)=e_α E_0(ωsin(ω_α
        t)-ω_αsin(ω t))/ω_α(ω^2-ω_α^2).
    
    As for the case without external time-dependent fields, we find that our
    F_P(t) is defined in terms of initial positions and momenta of bath
    oscillators, but in our case it now also depends on the sinusoidal electric
    field at time t. Note that, by shifting the time origin, it can be easily
    verified that the statistical average is stationary. Following
    Zwanzig <cit.>, we assume the initial conditions for the bath
    oscillators can be taken to be Boltzmann-distributed ∼exp(-H_B/k_BT), where the bath is in thermal equilibrium with respect to
    a frozen or constrained system coordinate X(0).
    
    Then for the average of X and P we find:
    
        ⟨ X_α(0)-c_α Q(0)/ω_α^2⟩=0,   ⟨ P_α(0)⟩=0.
    
    The second moments are
    
        ⟨(X_α(0)-c_α
        Q(0)/ω_α^2)^2⟩=k_BT/m_αω_α^2,
          ⟨ P_α(0)^2⟩=m_α k_BT.
    
    Both these results are consistent with what one finds for systems without
    external time-dependent fields, since obviously they descend directly from the
    assumption of Boltzmann-distributed degrees of freedom at the initial time.
    
    As for ∑_α c_α E_α(t), we first note that there is no
    singularity when ω_α→ω and
    ω_α→0:
    
        lim_ω_α→ωωsin(ω_α
        t)-ω_αsin(ω
        t)/ω_α(ω^2-ω_α^2)   =-ω tcos(ω t
        )-sin(ω t)/2ω^2
        lim_ω_α→0ωsin(ω_α
        t)-ω_αsin(ω
        t)/ω_α(ω^2-ω_α^2)   =ω t-sin(ω
        t)/ω^2.
    
    
    
    Focusing on ions, atoms or molecules, ω_α is at least in the THz
    regime or much higher, which is orders of magnitude larger than the frequency of the
    applied field ω. Hence, the
    first term in the numerator in the RHS of Eq. (8) can be neglected, and we can
    approximate as follows:
    
        ∑_α c_α E_α(t)≈∑_αc_α
        e_α/ω_α^2E_0sin(ω t) eE(t).
    
    
    We now take the ensemble average of the stochastic force, Eq. (7), and we
    find:
    
        ⟨ F_P(t)⟩=γ eE(t),
    
    for some constant γ.
    That is, the average of the stochastic force F_P is non-zero, unlike in all
    previously studied GLEs of particle-bath systems and, remarkably, is directly
    proportional to the AC field.
    
    
    
    
    Now, by direct calculation, using Eq. (10) and standard trigonometric
    identities, we can get the fluctuation-disspation theorem (FDT) for our
    particle-bath Hamiltonian under a uniform AC field:
    
        ⟨ F_P(t)F_P(t')⟩   =1/Z_N∫
        F_P(t)F_P(t')exp(-H_B/k_BT)d𝐗(0)
        d𝐏(0)
           =∑_α(m_α
        c_α^2k_BT/ω_α^2cos(ω_α
        t)cos(ω_α t')+m_α
        c_α^2k_BT/ω_α^2sin(ω_α
        t)sin(ω_α t'))+(γ e)^2E(t)E(t')
           =k_BT∑_αm_α
        c_α^2/ω_α^2cos(ω_α(t-t'))+(γ
        e)^2E(t)E(t')
           =mk_BTν(t-t')+(γ e)^2E(t)E(t')
    
    where Z_N is the canonical partition function
    
        Z_N=∫exp(-H_B/k_BT)d𝐗(0)d𝐏(0)
    
    and 𝐗(0)={X_1(0),X_2(0),...}, 𝐏(0)={P_1(0),P_2(0),...}.
    
    Equation (14) is a key result of this paper and is the FDT associated with the
    GLE given by Eq. (6).
    This is a remarkable result which shows that in the presence of an external AC
    field which affects the microscopic dynamics of both the tagged particle and
    the bath oscillators, the strength of the noise is no longer proportional to
    thermal energy only, but also has an important non-thermal contribution
    proportional to the AC field amplitude squared.
    
    The implications of this result for statistical mechanics and its applications
    are broad and will be explored in future works,
    including experimental verifications.
    We can however notice that the presence of the external AC field makes it
    impossible for the noise to be white noise. This is true even if the memory
    function is delta-correlated, because of the second term controlled by the AC
    field in Eq. (14), which inevitably introduces a deterministic
    temperature-independent contribution to the noise.
    
    A further consideration is about the average of the noise, expressed
    by Eq. (13). Unlike all previously derived versions of the GLE, the stochastic
    force here does not have zero average but is instead directly proportional to
    the AC electric field.
    
    In conclusion, we have introduced a more general version of the classical
    particle-bath Hamiltonian, which is used as a starting point to derive
    Generalized Langevin Equations, for systems subject to an external
    time-dependent (oscillating) field. Unlike in previous models where the bath
    oscillators were always taken to be unaffected by the field, here we added the
    time-dependent force due to the field to both the Hamiltonian of the particle
    and the Hamiltonian of the bath oscillators. The resulting Hamiltonian has been
    solved analytically and the resulting GLE and fluctuation-dissipation theorem
    have been found. The structure of the GLE is still formally similar to
    that of standard GLE with external field acting on the particle
    only <cit.> (and the memory function for the friction is the same), but
    the stochastic force is very different. Its ensemble average, remarkably, is
    non-zero and directly proportional to the AC field. The associated
    fluctuation-dissipation theorem has an additional term given by the
    time-correlation of the AC field, and is thus quadratic in the field amplitude.
    
    An immediate application of this result is to dielectric relaxation and
    dielectric spectroscopy of liquids and glasses. The Debye model treats each
    molecule as fully independent from the other molecules in the material and
    describes it with a Langevin equation for the orientation of the molecule in
    the field <cit.>. More refined models, e.g. Mode-Coupling Theory (MCT),
    are able to account for the coupling of each molecule to collective degrees of
    freedom <cit.>, but do not explicitly address the simultaneous effect of the
    AC field on the dynamics of both the single molecule and all the other molecules
    which provide the memory effect in dielectric relaxation. The GLE derived here
    will open the possibility of describing both these effects at the same time,
    within MCT type approaches <cit.> or within the GLE picture that has been proposed more recently to deal with
    hierarchy of relaxation times in liquids and glasses <cit.>.
    Furthermore, the GLE derived here can be used as the starting point for a more
    microscopic description of nonlinear dielectric response of
    supercooled liquids under strong fields <cit.>, for which a microscopic
    picture is currently lacking <cit.>. Further applications of the proposed framework include quantum dissipative
    transport and Josephson tunnelling with dissipation <cit.>,
    driven dynamics of colloids in soft matter systems <cit.>, and molecular dynamics simulations of
    liquids <cit.> and of amorphous solids in oscillatory
    shear <cit.>.
    
    
    
    B. Cui acknowledges financial support from the CSC-Cambridge scholarship.
    We had useful discussions with Gerhard Stock.
    
    
    99Onsager
    L. Onsager, Reciprocal relations in irreversible processes I., Phys.
    Rev. 37, 405-426 (1931).
    
    
    Callen
    H. B. Callen and T. A. Welton, Irrevisibility and generalized noise,
    Phys. Rev. 83, 34-40 (1951),
    
    Leontovich
    M. A. Leontovich and S. M. Rytov, Electric fluctuations, J. Exper.
    Theor. Phy. USSR 23, 246-252 (1952).
    
    Rytov
    S. M. Rytov, Theory of electrical fluctuations, Academy of
    Sciences USSR. Moscow (Publishing House 1953).
    
    Kubo
    R. Kubo, Statistical-mechanical theory of irreversible
    processes I, J. Phys. Soc. Japan 12, 570-286 (1957).
    
    Bernard
    W. Bernard and H. B. Callen, Irreversible thermodynamics
    of nonlinear processes and noise, Rev. Mod. Phys. 31, 1017-1044 (1959).
    
    Akhiezer
    A. I. Akhiezer, I. A. Akhiezer, R. V. Polivin, et al, Collective
    oscillations in a plasma, (Oxford: Pergamon Press; 1967).
    
    Landau
    L. D. Landau and E. M. Lifshitz, Statistical physics, (Oxford:
    Pergamon Press; 1959).
    
    Seifert1
    U. Seifert, T. Speck, Fluctuation-dissipation theorem in nonequilibrium
    steady states
     Phys. Rev. Lett. 89, 10007 (2010).
    
    Seifert2
    V. Blickle, T. Speck, L. Helden, U. Seifert, and C. Bechinger,
    Thermodynamics of a Colloidal Particle in a Time-Dependent Nonharmonic
    Potential Phys. Rev. Lett. 96, 070603 (2006).
    
    Rubi
    A. Perez-Madrid, D. Reguera, J. M. Rubi, Origin of the violation of the
    fluctuationâdissipation
    theorem in systems with activated dynamics, Physica A 329, 357-364 (2003).
    
    Ford
    G. W. Ford, The fluctuation-disspation theorem, Contemp. Phys. 58:3,
    244-252 (2017).
    
    Haenggi
    P. Haenggi and G.L. Ingold, Fundamental aspects of quantum Brownian motion, Chaos 15, 026105 (2005).
    
    Zwanzig
    R. Zwanzig, Nonequilbrium Statistical Mechanics (Oxford University Press, New
    York, 2001), R. Zwanzig, Nonlinear generalized Langevin equations, J. Stat.
    Phys. 9, 215 (1973).
    
    Debye
    P. Debye. Polar Molecules (The Chemical Catalog Inc., New York, 1929).
    
    Coffey
    W. T. Coffey, Yu. P. Kalmykov, J. T. Waldron, The Langevin Equation (World
    Scientific, Singapore, 2004).
    
    Goetze
    W. Goetze and T. Voigtmann, Universal and nonuniversal features of
    glassy relaxation in propylene carbonate. Phys. Rev. E 61, 4133-4147 (2000);
    W. Gotze, A.P. Singh, T. Voigtmann, Reorientational relaxation of a
    linear probe molecule in a simple glassy liquid. Phys. Rev. E 61, 6934-6949
    (2000); M. Sperl, Cole-Cole law for critical dynamics in glass-forming liquids, Phys. Rev. E 74, 011503 (2006).
    
    Blochowitz
    M. Domschke, M. Marsilius, T. Blochowitz, and T. Voigtmann, Glassy
    relaxation and excess wing in mode-coupling theory: The dynamic susceptibility
    of propylene carbonate above and below Tc. Phys. Rev. 84, 031506 (2011).
    
    Rubi1
    L. C. Lapas, A. Perez-Madrid, and M. J. Rubi, Theory of Casimir Forces
    without the Proximity-Force Approximation. Phys. Rev. Lett. 116, 110601
    (2016).
    
    Rubi2
    A. Perez-Madrid, L. C. Lapas, and J. M. Rubi, Multiscale model for
    dielectric permittivity. Z. Naturforsch. 72, 109 (2017).
    
    Cui
    B. Cui, R. Milkus, and A. Zaccone, Direct link between boson-peak modes
    and dielectric α-relaxation in glasses. Phys. Rev. E 95, 022603 (2017).
    
    Loidl
    S. Albert, et al., Fifth-order susceptibility unveils growth of
    thermodynamic amorphous order in glass-formers. Science
    352, 1308 (2016); T. Bauer, P. Lunkenheimer, S. Kastner and A. Loidl,
    Nonlinear dielectric response at the excess wing of glassforming
    liquids. Phys. Rev. Lett. 110, 107603 (2013).
    
    Richert
    R. Richert, Nonlinear dielectric effects in liquids:
    a guided tour. J. Phys.: Condens. Matter 29, 363001 (2017).
    
    Breuer
    F. Petruccione and H.-P. Breuer, The Theory of Open Quantum Systems (Oxford
    University Press, Oxford, 2002).
    
    Caldeira
    A. O. Caldeira, An Introduction to Macroscopic Quantum Phenomena and Quantum
    Dissipation (Cambridge University Press, Cambridge, 2014).
    
    Voigtmann
    A.M. Puertas and T. Voigtmann, Microrheology of colloidal systems, J.
    Phys. Condens. Matter  26,  243101 (2014).
    
    Barrat
    P. Ilg and J.-L. Barrat, Driven vs. Thermal Activation, EPL 79,  26001
    (2007).
    
    Yurchenko
    E. V. Yakovlev, et al., Tunable two-dimensional
    assembly
    of colloidal particles in rotating
    electric fields. Sci. Rep. 7, 13727 (2017).
    
    Schilling
    H. Meyer, Th. Voigtmann, T. Schilling
    On the non-stationary Generalized Langevin Equation
    J. Chem. Phys. 147, 214110 (2017).
    
    Damart
    T. Damart, A. Tanguy, and D. Rodney, Theory of harmonic dissipation in
    disordered solids. Phys. Rev. B 95, 054203 (2017).
