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Statistical ensembles and probability density

Dans le document The DART-Europe E-theses Portal (Page 27-32)

Molecular Dynamics simulation methods

1.2 Statistical ensembles and probability density

In this thesis, the numerical methods and concepts implemented are based on statistical physics. This is why a few reminders about statistical physics are presented here. A short summary is provided but several references can also support the calculations [130]. The general idea of statistical physics is to describe macroscopic properties from the behaviour of individual molecules composing the system. It is impossible of course to describe the movement of each molecule or each atom of a macroscopic system, which justifies the statis-tical approach. In this frame, the thermodynamic quantities must be defined, such as the pressure and the temperature, and the laws of thermodynamics are

-1 -0.5 0 0.5 1 Position

-1 -0.5 0 0.5 1

Velocity

Figure 1.1: Example of phase space. This curve represents the motion of a one-dimensional damped oscillator.

derived from microscopic and statistical descriptions.

1.2.1 Probability density

From a classical point of view, a system of N particles is entirely described by the position and impulsion of each particle consisting in 6N components, i.e.

a point in the phase space. The position of this point may depend on time and thus creates a curve on the phase diagram, as shown in Figure 1.1 for a one-dimensional damped oscillator The macroscopic state of this system results from this microscopic state. However, they are several microscopic states that are macroscopically identical. Those states are explored during the fluctuations of the system. If one could measure the position and impulsion of each particle of a large collection of identical systems, an histogram of all states could be created and the probability law of the system could be deduced. The average defined by this probability law is called the ensemble average and the collection of systems is called statistical ensemble.

To build the probability density, one can imagine measuring the positions and impulsions of all particles of the same system but at different times. The average using this density probability is a temporal average and it is not always equal to the ensemble average. When they are equal the system is said to be ergodic but all systems are not ergodic. An example of non-ergodic system is given in AnnexeA.

1.2.2 Microcanonical ensemble (NVE)

All the quantities defined in the frame of thermodynamics can be defined in statistical physics as an ensemble average, but this description requires the knowledge of the probability density of the states of the system. To address this problem, let us consider an isolated system at equilibrium. Its total energy is a constant and it determines the accessible states of the systems. The collection of the accessible states of an isolated system is called microcanonical ensemble sometimes noted NVE, for fixed number of particles N, fixed volume V and fixed energy E.

For an isolated system, it is postulated that all its accessible states have the same associated probability. This postulate is justify by the results of the theory based on this postulate and its agreement with experiments. Using this postulate, the usual thermodynamic quantities can be derived.

1.2.3 Entropy

Entropy has been first introduced as a thermodynamics quantity in the second law of thermodynamics and later Ludwig Boltzmann introduced an expression of the entropy,S, based on the probability density. It was then reformulated by Shannon in the information theory as follows, withkBthe Boltzmann constant:

S =−kBX

i

PilnPi. (1.1)

However, this expression and whether the second law of physics only emerges from statistical physics or not is still debated specially concerning non-equilibrium statistical mechanics and irreversible processes [149,31,103]. This question will not be debated here and the Boltzmann expression will be used.

When the probability of each state is the same, as in the microcanonical en-semble, Pi = 1,Ωbeing the number of states yielding the entropy becomes:

S = −kBX

The second principle of statistical physics states that the equilibrium probabil-ity densprobabil-ity is the one that maximizes the entropy of the system.

1.2.4 Canonical ensemble (NVT)

Let us study a system that is not isolated but in contact with a thermal bath.

It can exchange heat with this latter reservoir, as a result, its energy fluctuates but at equilibrium, its average value is fixed. This system creates a canonical

ensemble, noted NVT, for fixed number of particles N, fixed volume V and fixed

withEi is the energy of state iand hEi is the average energy of the system.

To find the probability law, the Lagrange multiplier methods can be used by introducing the multipliersλ1 and λ2 to find the maximum of the entropy:

−kB

The derivation ofPk leads to

−kB(1 + lnPk) +λ12Ek= 0, (1.6) thus,

Pi =e−1+

λ1+λ2Ei

kB , (1.7)

which can be restated as

Pi= e−βEi

Z , (1.8)

withZ the partition function

Z=X

i

e−βEi, (1.9)

andβ is called the statistical temperature.

Using this probability lawPi, hEi=X

β can now be identified by estimating the entropyS:

S =−kBX

i

PilnPi=−kBlnZ+kBβhEi. (1.14)

Consequently,

∂S

∂hEi =kBβ. (1.15)

Finally, by identification with the thermodynamic expression, β = 1

kBT. (1.16)

1.2.5 NPT ensemble

In most cases studied in this manuscript, the systems do not have a fixed volume as they are in contact with a constant pressure bath, this systems are noted NPT. The probability law in this ensemble can be obtained with the same procedure as in NVT ensemble and adding the condition of fixed pressure Pr.

X

Let us find the probability law by deriving the following expression with respect to Pk:

Leading to the probability law:

Pi= e−βEi2Vi λ2 is identified using

T ∂S

∂hVi = P r,

= −λ2kBT.

This relation defines the pressure P r. Finally, using the enthalpy, H = E + VPr, the probability law is obtained

Pi = e−βH

Z . (1.23)

This expression is thus very similar to the one of the canonical ensemble. These relations can be generalized to different quantities, in particular when the num-ber of particlesN can vary, this ensemble is called grand canonical ensemble.

Dans le document The DART-Europe E-theses Portal (Page 27-32)