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Nano Mechanics and Materials: Theory, Multiscale Methods and Applications by Wing Kam Liu, Eduard G. Karpov, Harold S. Park

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4. Methods of Thermodynamics and Statistical Mechanics

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Basic features of the thermodynamic method: Multi-particle physical systems is described by means of a small number of macroscopically measurable parameters, the thermodynamic parameters: V, P, T, S (volume, pressure, temperature, entropy), and others. Note: macroscopic objects contain ~ 10 23 …10 24 atoms (Avogadro’s number ~ 6 x 10 23 mol –1 ). The connections between thermodynamic parameters are found from the general laws of thermodynamics. The laws of thermodynamics are regarded as experimental facts. Therefore, thermodynamics is a phenomenological theory. Thermodynamics is in fact a theory of equilibrium states, i.e. the states with time- independent (relaxed) V, P, T and S. Term “dynamics” is understood only in the sense “how one thermodynamic parameters varies with a change of another parameter in two successive equilibrium states of the system”. Thermodynamics is a macroscopic, phenomenological theory of heat. Definition and Features the Thermodynamic Method

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Internal and external parameters: External parameters can be prescribed by means of external influences on the system by specifying external boundaries and fields. Internal parameters are determined by the state of the system itself for given values of the external parameters. Note: the same parameter may appear as external in one system, and as internal in another system. Intensive and extensive parameters: Intensive parameters are independent of the number of particles in the system, and they serve as general characteristics of the thermal atomic motion (temperature, chemical potential). Extensive parameters are proportional to the total mass or the number of particles in the system (internal energy, entropy). Note: this classification is invariant with respect to the choice of a system. Classification of Thermodynamic Parameters

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A same parameter may appear both as external and internal in various systems: System A System B P V = Const M P = Const V External parameter: V External parameter: P, P = Mg/A Internal parameter: P Internal parameter: V, V = Ah Internal and External Parameters: Examples

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Application of the thermodynamic method implies that the system if found in the state of thermodynamic equilibrium, denoted X, which is defined by time-invariant state parameters, such as volume, temperature and pressure: The parameters (V,T,P) are macroscopically measurable. One or two of them may be replaced by non-measurable parameters, such internal energy or entropy. Note that only the mean quantity of a state parameter A is time-invariant, see the plot. A mathematical relationship that involve a complete set of measurable parameters (V,T,P) is called the thermodynamic state equation Here, ξ is the vector of system parameters State Vector and State Equation

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The knowledge of an equation of state allows evaluation of a group of the microscopic system parameters, such as the compressibility, expansion and pressure coefficients: 1) Isothermal compressibility coefficient 2) Isobaric thermal expansion coefficient 3) Isochoric pressure coefficient Analysis of the State Equation: System Parameters

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Standard forms of the state equations are readily available for ideal gases, real (Van der Waals) gases, homogeneous liquid and homogeneous isotropic solids. Ideal gas Van der Waals gas (pair wise interaction V of gas molecules is taken into account) Homogeneous isotropic liquid or solid d – effective diameter of the molecules W – pairwise potential Examples of the State Equation

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The temperature is introduced as a parameter, which: 1) serves as an intrinsic characteristic of any equilibrium system (similar to V and P) 2) determines thermodynamic equilibrium between two systems in thermal contact Thus, it is postulated that: If two adiabatically isolated systems in equilibrium are brought into thermal contact with each other, their states of equilibrium will not be altered and the total system will be in equilibrium, only if initial systems have the same temperature. (Also known as the zeroth law of thermodynamics) Any state of thermodynamic equilibrium of an arbitrary system in entirely determined by the set of external parameters and temperature. Consequently, All internal parameters of an equilibrium system are functions of the external parameters and temperature. The Postulate on Existence of Temperature

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The first law of thermodynamics is essentially a form of the energy conservation law, written in relation to thermodynamic systems: An amount of heat absorbed by the system is equal to the summary change of its internal energy and the work done by the system over external bodies. Note: The internal energy U is defined solely by the state of system, while the external thermal energy Q and the mechanical work W may depend both on the internal state of the system and other factors. The First Law of Thermodynamics

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The second law of thermodynamics specifies the direction of thermodynamic processes. The simplest form of the second law is given by Clausius’ postulate: Heat cannot flow spontaneously from a colder to a hotter system. This is equivalent to the following (Kelvin’s postulate): It is impossible to devise an engine (a perpetuum mobile of the second kind) which, working in a cycle, would produce no other effect than the transformation of heat extracted from a reservoir completely into work. The Second Law of Thermodynamics

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The most important consequence of the second law of thermodynamics is that it asserts the existence of a new function of state, namely the entropy, S. The concept of entropy plays a crucial role in statistical mechanics. For quasistatic processes, entropy is an extensive state function, which is defined by the relation Based on the first law of thermodynamics we obtain the fundamental differential equation Alternative form of the second law of thermodynamics: All spontaneous processes in adiabatically isolated systems, δQ = 0, occur at a constant or growing entropy: Entropy

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The second law and the original definition of entropy does not specify the absolute value of entropy, whose value is provided up to a constant value. The third law claims that All thermodynamic processes at T = 0 occur without a change of the entropy. This allows establishing an absolute scale for measuring the entropy; so that Also, The Third Law of Thermodynamics

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The method of thermodynamic potentials is a powerful tool of thermodynamics. Thermodynamic potentials are functions that uniquely describe the state of the system. The relevant thermodynamic parameters are found as derivatives of the potentials. The simplest thermodynamic potential is the internal energy, given by the first law, Other thermodynamic potentials for systems with constant number of particles: free energy Gibbs potential enthalpy Exercise 3-1 (can be done in class): derive the differentials of the free energy, Gibbs potential and enthalpy, by utilizing the definitions of these potentials and the first law of thermodynamics. Thermodynamic Potentials

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4.1 Basic Results of the Thermodynamic Method Differentiation of the thermodynamic potentials gives the thermodynamic parameters: Also, system parameters can be evaluated using their relationship with the above state parameters (see reading assignment [1], Section 4.1.5)

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Hamiltonian of the system and the generalized momentum: Equations of motion: Cartesian coordinates: Hamiltonian Mechanics

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In classical (deterministic) mechanics, the state of the system is completely described by the phase vector Q. Such a state is called a microscopic state, and the corresponding model of matter is called a micromodel. The microstate is completely defined by specifying values of all canonical variables, the components of the phase vector, This approach is not tractable in modeling macroscopic objects. Thermodynamics provides a macromodel; the state of the a system is determined by a very limited number of thermodynamic parameters, which are sufficient for macroscopic characterization of the system. The prescription of these parameters, measured in a macroscopic experiment, determines the macroscopic state the system. A key point is that a single macroscopic state of the system corresponds to a great number of different microscopic states. Micromodel vs. Macromodel

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4.2 Statistics of Multiparticle Systems in Thermodynamic Equilibrium However, the specification of all the macroparameters X does not determine a unique microstate, Consequently, on the basis of macroscopic measurements, one can make only statistical statements about the values of the microscopic variables. The macroscopic thermodynamic parameters, X = (V,P,T,…), are macroscopically observable quantities that are, in principle, functions of the canonical variables, i.e.

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Statistical description of mechanical systems is utilized for multi-particle problems, where individual solutions for all the constitutive atoms are not affordable, or necessary. Statistical description can be used to reproduce averaged macroscopic parameters and properties of the system. Comparison of objectives of the deterministic and statistical approaches: Deterministic particle dynamicsStatistical mechanics Provides the phase vector, as a function of time Q(t), based on the vector of initial conditions Q(0) Provides the time-dependent probability density to observe the phase vector Q, w(Q,t), based on the initial value w(Q,0) Statistical Description of Mechanical Systems

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From the contemporary point of view, statistical mechanics can be regarded as a hierarchical multiscale method, which eliminates the atomistic degrees of freedom, while establishing a deterministic mapping from the atomic to macroscale variables, and a probabilistic mapping from the macroscale to the atomic variables: Microstates Macrostates deterministic conformity probabilistic conformity XkXk (p,q) k Statistical Description of Mechanical Systems

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Though the specification of a macrostate X i cannot determine the microstate (p,q) i = (p 1,p 2,…,p s ; q 1,q 2,…,q s ) i, a probability density w of all the microstates can be found, or abbreviated: The probability of finding the system in a given phase volume G: The normalization condition: Distribution Function

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Within the statistical description, the motion of one single system with given initial conditions is not considered; thus, p(t), q(t) are not sought. Instead, the motion of a whole set of phase points, representing the collection of possible states of the given system. Such a set of phase points is called a phase space ensemble. If each point in the phase space is considered as a random quantity with a particular probability ascribed to every possible state (i.e. a probability density w(p,q,t) is introduced in the phase space), the relevant phase space ensemble is called a statistical ensemble. G – volume in the phase space, occupied by the statistical ensemble. p q t = t 1 : G 1 t = t 2 : G 2 Statistical Ensemble

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Statistical average (expectation) of an arbitrary physical quantity F(p,q), is given most generally by the ensemble average, The root-mean-square fluctuation (standard deviation): The curve representing the real motion (the experimental curve) will mostly proceed within the band of width 2Δ(F) For some standard equilibrium systems, thermodynamic parameters can be obtained, using a single phase space integral. This approach is discussed below. True value F t Statistical Averaging

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Evaluation of the ensemble average (previous slide) requires the knowledge of the distribution function w for a system of interest. Alternatively, the statistical average can be obtained by utilizing the ergodic hypothesis in the form, Here, the right-hand side is the time average (in practice, time t is chosen finite, though as large as possible) This approach requires F as a function of the generalized coordinates. Some examples More examples are given in Ref. [1]. Ergodic Hypothesis and the Time Average

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A statistical ensemble is described by the probability density in phase space, w(p,q,t). It is important to know how to find w(p,q,t) at an arbitrary time t, when the initial function w(p,q,0) at the time t = 0 is given. In other words, the equation of motion satisfied by the function w(p,q,t) is needed. p q w(p,q,0) Γ2Γ2 Γ0Γ0 Γ1Γ1 w(p,q,t 1 ) w(p,q,t 2 ) The motion of of an ensemble in phase space may be considered as the motion of a phase space fluid in analogy to the motion of an ordinary fluid in a 3D space. Liouville’s theorem claims that Due to Liouville’s theorem, the following equation of motion holds Law of Motion of a Statistical Ensemble

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Exercise: Check the above equality. A direct solution of this equation is not tractable. Therefore, the ergodic hypothesis (in a more general form) is utilized: the probability density in phase space at equilibrium depends only on the total energy: Notes: the Hamiltonian gives the total energy required; the Hamiltonian may depend on the values of external parameters a = (a 1, a 2,…), besides the phase vector X. This distribution function satisfies the equilibrium equation of motion, because For a system in a state of thermodynamic equilibrium the probability density in phase space must not depend explicitly on time, Thus, the equation of motion for an equilibrium statistical ensemble reads Equilibrium Statistical Ensemble: Ergodic Hypothesis

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After adoption of the ergodic hypothesis, it then remains to determine the actual form of the function φ(H). This function depends on the type of the thermodynamic system under consideration, i.e. on the character of the interaction between the system and the external bodies. We will consider canonical ensembles of two types of systems: 1) Adiabatically isolated systems that have no contact with the surroundings and have a specified energy E. The corresponding statistical ensemble is referred to as the microcanonical ensemble, and the distribution function – microcanonical distribution. 2) Closed isothermal systems that are in contact and thermal equilibrium with an external thermostat of a given temperature T. The corresponding statistical ensemble is referred to as the canonical ensemble, and the distribution function – Gibbs’ canonical distribution. Both systems do not exchange particles with the environment. Canonical Ensembles

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For an adiabatically isolated system with constant external parameters, a, the total energy cannot vary. Therefore, only such microstates X can occur, for which This implies (δ – Dirac’s delta function) and finally: where Ω is the normalization factor, E, a (P,T,V,…) Within the microcanonical ensemble, all the energetically allowed microstates have an equal probability to occur. Microcanonical Distribution

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The normalization factor Ω is given by where Γ is the integral over states, or phase integral: Γ(E,a) represents the normalized phase volume, enclosed within the hypersurface of given energy determined by the equation H(X,a) = E. Phase integral Γ is a dimensionless quantity. Thus the normalization factor Ω shows the rate at which the phase volume varies due to a change of total energy at fixed external parameters. Phase volume Microcanonical Distribution: Integral Over States

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The integral over states is a major calculation characteristic of the microcanonical ensemble. The knowledge of Γ allows computing thermodynamic parameters of the closed adiabatic system: (These are the major results in terms of practical calculations over microcanonical ensembles.) Microcanonical Distribution: Integral Over States

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We will consider one-dimensional illustrative examples of computing the phase integral, entropy and temperature for microcanonical ensembles: Spring-mass harmonic oscillator Pendulum (non-harmonic oscillator) We will use the Hamiltonian equations of motion to get the phase space trajectory, and then evaluate the phase integral. Microcanonical Ensemble: Illustrative Examples

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Hamiltonian: general form Kinetic energy Potential energy The total Hamiltonian Potential energy is a quadratic function of the coordinate (displacement form the equilibrium position) m k x Harmonic Oscillator: Hamiltonian

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Hamiltonian and equations motion: Initial conditions (m, m/s): Harmonic Oscillator: Equations of Motion and Solution

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Total energy: Harmonic Oscillator: Total Energy

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Phase integral: A2A2 A3A3 A1A1 For the harmonic oscillator, phase volume grows linearly with the increase of total energy. Harmonic Oscillator: Phase Integral

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Entropy: Temperature: We perturb the initial conditions (on 0.1% or less) and compute new values The temperature is computed then, as Harmonic Oscillator: Entropy and Temperature

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Total energy: Pendulum: Total Energy

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Phase integral: For the pendulum, phase volume grows NON-linearly with the increase of total energy at large amplitudes. A2A2 A3A3 A1A1 Pendulum: Phase Integral

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Entropy: Temperature: We perturb the initial conditions (on 0.1% or less) and compute new values The temperature is computed then, as Pendulum: Entropy and Temperature

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1.Analyze the physical model; justify applicability of the microcanonical distribution. 2.Model individual particles and boundaries. 3.Model interaction between particles and between particles and boundaries. 4.Set up initial conditions and solve for the deterministic trajectories (MD). 5.Compute two values of the total energy and the phase integral – for the original and perturbed initial conditions. 6.Using the method of thermodynamic parameters, compute entropy, temperature and other thermodynamic parameters. If possible compare the obtained value of temperature with benchmark values. 7.If required, accomplish an extended analysis of macroscopic properties (e.g. functions T(E), S(E), S(T), etc.) by repeating the steps 4-7. Summary of the Statistical Method: Microcanonical Distribution

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Canonical Distribution: Preliminary Issues One important preliminary issue related to the use of Gibbs’ canonical distribution is the additivity of the Hamiltonian of a mechanical system. Structure of the Hamiltonian of an atomic system: Here, kinetic energy and the one-body potential are additive, i.e. they can be expanded into the components, each corresponding to one particle in the system: Two-body and higher order potentials are non-additive (function Q 2 does not exist),

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Canonical Distribution: Preliminary Issues Thus, if the inter-particle interaction is negligible, the system is described by an additive Hamiltonian, Here, H is the total Hamiltonian, and h i is the one-particle Hamiltonian. For the statistical description, it is sufficient that this requirement holds for the averaged quantities only. The multi-body components, W >1, cannot be completely excluded from the physical consideration, as they are responsible for heat transfer and establishing the thermodynamic equilibrium between constitutive parts of the total system. A micromodel with small averaged contributions to the total energy due to particle-particle interactions is called the ideal gas. Example: particles in a circular cavity. Statistically averaged value W 2 is small:

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Canonical Distribution Suppose that system under investigation Σ 1 is in thermal contact and thermal equilibrium with a much larger system Σ 2 that serve as the thermostat, or “heat bath” at the temperature T. From the microscopic point of view, both Σ 1 and Σ 2 are mechanical systems whose states are described by the phase vectors (sets of canonical variables X 1 and X 2 ). The entire system Σ 1 +Σ 2 is adiabatically isolated, and therefore the microcanonical distribution is applicable to Σ 1 +Σ 2, Assume N 1 and N 2 are number of particles in Σ 1 and Σ 2 respectively. Provided that N 1 << N 2, the Gibbs’ canonical distribution applies to Σ 1 : Thermostat T N 1 Σ 1 N 2 Σ 2

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Canonical Distribution: Partition Function The normalization factor Z for the canonical distribution called the integral over states or partition function is computed as Thermostat T N 1 Σ 1 N 2 Σ 2 Before the normalization, this integral represents the statistically averaged phase volume occupied by the canonical ensemble. The total energy for the canonical ensemble is not fixed, and, in principle, it may occur arbitrary in the range from – to (for the infinitely large thermostat, N 2 ).

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Partition Function and Thermodynamic Properties The partition function Z is the major computational characteristic of the canonical ensemble. The knowledge of Z allows computing thermodynamic parameters of the closed isothermal system (a V, external parameter): These are the major results in terms of practical calculations over canonical ensembles. Class exercise: check the last three above formulas with the the method of thermodynamic potentials, using the first formula for the free energy. Free energy: (relates to mechanical work) Entropy (variety of microstates) Pressure Internal energy

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Free Energy and Isothermal Processes Free energy, also Helmholtz potential is of importance for the description of isothermal processes. It is defined as the difference between internal energy and the product of temperature and entropy. Since free energy is a thermodynamic potential, the function F(T,V,N,…) guarantees the full knowledge of all thermodynamic quantities. Physical content of free energy: the change of the free energy dF of a system at constant temperature, represents the work accomplished by, or over, the system. Indeed, Isothermal processes tend to a minimum of free energy, i.e. due to the definition, simultaneously to a minimum of internal energy and maximum of entropy.

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Canonical vs. Microcanonical: Factorization of the Partition Function In terms of practical calculations, there exists one major difference between the canonical and microcanonical distributions: For additive Hamiltonians, the canonical distribution factorizes, Note that this property does not hold for the microcanonical distribution, Thermostat T 6 N 1 Σ 1 N 2 Σ 2 E, a z i is the one-particle partition function

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Factorization of the Partition Function: Computational Issues Factorization of the canonical distribution is crucial in terms of practical calculations over real-life systems. In computing the partition function Z, this property reduces the calculation of a 6N- dimensional phase integral to a product of N 6-dimensional integrals: Here, z i is the one-particle partition function, and h i is the one-particle Hamiltonian, In case that the system is comprised of identical particles, calculation of Z requires evaluation of a single 6-dimensional integral z: The factorized canonical distribution can be very effective computationally.

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Analytical Example: Non-Interactive Ideal Gas The canonical distribution allows an exact solution for the non-interactive ideal gas: Analytical results for this system are useful, because they provide acceptable “first guess” assessments for a wide class of systems. One-particle partition function: Partition function (total system):

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Non-Interactive Ideal Gas: Thermodynamic Parameters Partition function: Free energy (recall the method of thermodynamic potentials) Entropy Pressure Total internal energy (differs form the earlier MD definition)

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Numerical Example: Interactive Gas Repulsive interaction between the particles and the wall is described by the “wall function”, a one-body potential that depends on r i – distance between the particle i and the chamber’s center): Interaction between particles is modeled with the two-body Lennard-Jones potential (r ij – distance between particles i and j): The Hamiltonian: y x riri R r ij rjrj One particle is initially at rest. This illustrates the concept of heat exchange between the smaller subsystem, for which the canonical distribution holds, and the external thermostat.

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Interactive Gas: Equations of Motion and Solution The total potential: Equations of motion: Parameters: Initial conditions (nm, nm/s):

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Interactive Gas: Temperature Time averaged kinetic energy vs. time (five particles) Information on temperature allows computing the partition function (integral over states), using canonical distribution. Subsequently, the partition function, computed at various temperatures, can provide all the remaining thermodynamic parameters. For sufficiently long simulations, the value of temperature does not depend on the choice of a subsystem (particle). Time averaged kinetic energy of particles is approaching the value which corresponds to temperature

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Interactive Gas: Partition Function and Free Energy Hamiltonian (can be viewed as additive in the statistical sense, due to smallness of the time averaged pair-wise interaction) One-particle partition function (value at given T and V = πR 2 ) Partition function (total system) Free energy

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Interactive Gas: Thermodynamic Parameters In order to compute the thermodynamic quantities, it necessary to evaluate 2 values of the partition function: 1) Z – for the initially computed temperature T, 2) – for a perturbed temperature T +ΔT (ΔT/T < 0.1%). Note: the simulation needs to be run once only (not two times). Entropy: Pressure: Internal energy: Ideal gas benchmark: Other parameters can be computed using the method of thermodynamic potentials

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Phase Integral, Free Energy and Entropy vs. Temperature Assume that we observe the same isothermal system at various temperatures of thermostat. The following trends are available: Partition function Free energy Entropy Partition function, and therefore the phase volume occupied by this canonical ensemble, grows exponentially vs. temperature. Free energy decreases linearly; the work done by the system does not depend on temperature. Entropy decays vs. temperature. Physical implication (according to the second law): temperature cannot grow spontaneously in an isothermal system, once thermal equilibrium with the thermostat is established.

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Specifics of Calculations for Liquids and Solids In liquids, the energy due to pair-wise interaction between particles is close to the kinetic energy (per particle). However, interaction, w ij, between separate constitutive parts (subdomains) i and j is still weak, if compared with the total kinetic energy of the smaller domain. Indeed, the kinetic energy depends on the subdomain volume, while w ij depends on the surface area. Therefore, the Hamiltonian can be expanded into h i – Hamiltonians of the sufficiently large subdomains. Partition function (z – partition functions for N identical subdomains, n – number of subdomain particles) For reasonably small subdomains, numerical evaluation of the liquid’s partition function can be effective. A similar approach is also applicable to solids. Example: H hihi i j Note: In case of large w ij, the micro- canonical distribution should be utilized.

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1.Analyze the physical model; justify applicability of the Gibbs’ canonical distribution. 2.Model individual particles and boundaries. 3.Model interaction between particles and between particles and boundaries. 4.Set up initial conditions and solve for the deterministic trajectories (MD). 5.Compute the averaged kinetic energy and temperature, or assume T given. 6.Based on the canonical distribution, compute two values of the partition function: for the original and perturbed temperatures. 7.Compute the free energy and other thermodynamic parameters, using the method of thermodynamic potentials. If possible compare the obtained value of internal energy with a benchmark value. 8.If required, accomplish an extended analysis of macroscopic properties (e.g. dependences P(T), P(N), S(T), etc.) by repeating the steps 4-7. Summary of the Statistical Method: Canonical Distribution

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4.3 Numerical Heat Bath Techniques Berendsen thermostat Adelman-Doll thermostatting GLE Phonon heat bath Time-history kernel and transform techniques Random force and lattice normal modes

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Finite Temperatures A heat bath technique is required to represent a peripheral region at finite temperatures Berendsen thermostat for a standard Langevin equation Berendsen et al., JCP 81(8), 1984

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Finite Temperatures Adelman-Doll’s thermostatting GLE for gas-solid interface Adelman, Doll et al., JCP 64(6), 1976 Almost exactly what we seek, however, the update is needed: gas/solid interface -> solid/solid interface

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Phonon Heat Bath Phonon heat bath represents energy exchange due to correlated motion of lattice atoms along an imaginary atomic/continuum (solid-solid) interface Phonon heat bath is a configurational method t Karpov, Liu, preprint. atom next to the interface

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Time History Kernel (THK) The time history kernel shows the dependence of dynamics in two adjacent cells. Any time history kernel is related to the response function. … -2 -1 0 1 2 … f(t)

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Elimination of Degrees of Freedom: T = 0 Equations for the DoF n>0 are no longer required. We have taken them into account implicitly. … -2 -1 0 1 2 … … … Domain of interest Eliminated degrees of freedom

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Bridging Scale at T = 0: Impedance Boundary Conditions MD degrees of freedom outside the localized domain are solved implicitly + Due to atomistic nature of the model, the structural impedance is evaluated computed at the molecular scale. The MD domain is too large to solve, so that we eliminate the MD degrees of freedom outside the localized domain of interest. Collective atomic behavior of in the bulk material is represented by an impedance force applied at the formal MD/continuum interface: FE + Reduced MD + Impedance BC MD FE

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Dynamic Response Function: 1D Illustration Assume first neighbor interaction only: … n-2 n-1 n n+1 n+2 … Displacements Velocities Illustration Transfer of a unit pulse due to collision ( movie ):

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Discrete Fourier Transform (DFT) Discrete convolution Discrete functional sequences DFT of infinite sequences p – wavenumber, a real value between – and DFT of periodic sequences Here, p – integer value between –N/2 and N/2

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Numerical Laplace Transform Inversion – Laguerre polynomials, – coefficients computed using F(s) Weeks algorithm (J Assoc Comp Machinery 13, 1966, p.419) Sin-series expansion (J Assoc Comp Machinery 23, 1976, p.89) For an odd function f gives Most numerical algorithms for the Laplace transform inversion utilize series decompositions of the sought originals f(t) in terms of functions whose Laplace transform is tabulated. The expansion coefficients are found numerically from F(s). Examples:

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Elimination of Degrees of Freedom at T > 0 Equations for the DoF n>0 are no longer required. We have taken them into account implicitly: - mechanical response is described by the THK - thermal contact is described by the randon force R … -2 -1 0 1 2 … … … Domain of interestHeat bath Compare with:

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Random Force Term ( g – the lattice response function ) Is there a more effective way to compute R(t) ?

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Gibbs Distribution and Lattice Hamiltonian Gibbs distribution Lattice Hamiltonian Normal modes decomposition

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Random Force Term via Lattice Normal Modes Random force Distribution of the normal amplitudes and phases

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