**2.1 Formalism of path integrals**

**2.1.2 Polymer isomorphism and PIMD**

As mentioned before, in order to calculate, via computer simulation, time indepen-dent equilibrium statistical properties using path integrals, we need to introduce the idea of the "classical isomorphism" [25]. We are still considering for the moment a single particle moving in one dimensional potential V(bx). We recall the expression of the discrete partition function (see equation2.30), which is a good approximation of the quantum canonical partition function if P is large enough:

Z_{P} =

mP
2πβ~^{2}

P /2Z

dx1· · ·dxP

exp (

−1

~

P

X

k=1

mP

2β~(xk+1−xk)^{2}+β~
P V(xk)

)

(2.31)

withx_{P}_{+1}=x1 and Z_{P} denotes that we are atP large but finite.

Equation2.31is the classical configurational canonical partition function of a cyclic polymer chain moving in the potential V(x) but at an effective temperature ofP T in phase space variables (x,p). The classical analogy can be made more complete by introducing a set of momenta via a Gaussian integration. Then we recast the P Gaussian prefactor as a set of Gaussian integrals over the variablep1,· · · , pP so

that they look like momenta conjugate tox1,· · · , xP:

~ which is called the chain frequency. In the exponential, we have replaced the prefactor by Gaussian integrals. In these Gaussian integrals, the physical mass is appearing. However, since the prefactor does not affect any equilibrium averages we are free to choose the mass mas we like, so not necessarly the physical mass of the system.

As explained before, once we write the partition function in this way, we make appear
the classical Hamiltonian of a cyclic chain polymer at an effective temperature of
P T, a chain frequency ofω_{P} in the potential V(x):

The identificaton of the partition function, in equation2.32with the one of a cyclic
polymer ofP points, leads to the so called classical isomorphism. We are going to
exploit the isomorphism between the classical and approximate (becauseP is finite)
quantum partition function because it presents many numerical advantages. The
classical isomorphism is illustrated in Figure 2.3. Because the cyclic polymer looks
like a necklace, we call very often theP points "beads". The coils represent the fact
that neighbor beads are connected via spring characterized byω_{P}. According to the
classical isomorphism, we can use classicle techniques such as Molecular Dynamics
(MD) or Monte Carlo (MC) to compute approximate quantum properties.

This is not the only advantage of the polymer isomorphism. Indeed, this iso-morphism is very useful to visualize and deal with quantum systems. Most of the time the quantum reality goes against our natural intuition and path integrals via the classical isomorphism can help us to visualize it. The idea is to replace one

### 1

Figure 2.3: Intuitive picture of the classical isomorphism withP = 7.

quantum atom by a necklace of many classicle particles.

If we look closely to the equation 2.32, we notice thatω_{P} = _{β}^{P}

~. As a consequence, ωP increases linearly (atP fixed) with the temperature which means that the springs of the polymer become stiffer with the temperature and so at high temperature the P beads are almost not moving relative to one another. In this case the polymer evolution can be reduced to one point and we are in the classical limit.

This argument can be made more precise by considering the root mean square radius of gyration∆ of our ring polymer as:

∆^{2}= 1
P

P

X

k=1

hx^{2}_{k}i_{f} (2.36)

Wherefwill be defined in the equation2.47and corresponds to a probability density.

Focusing on a free particle (V(x) = 0), we obtain [26]:

∆ = Λ(T)

√8π (2.37)

with

Λ(T) =

2π~^{2}
mkT

1/2

(2.38)

High T Low T

1

Figure 2.4: Evolution of the ring polymer with the temperature.

Λ is the de Broglie thermal wave lenght. Thus the polymer collapses in one point if we increase the temperature. On the contrary, at low temperature the polymer is spreading more and more when the temperature decreases.

In the presence of a potential the spreading is limited by the potential itself. A poly-mer ring within a real system simulation experiences two opposing forces. From one side, the beads within an atom want to approach each other, since there are springs connecting them but, on the other hand, there is an inter atomic potential attracting beads from different atoms and spreading the necklaces. The thermal equilibrium is reached when the mean value of these two forces balances. However, should the number of beads per atom be not sufficiently high, the polymer would not spread as much as required for the given conditions of mass, temperature and potential. The results obtained in that situation would be biased towards the classical limit. But, if we put a sufficient amount of beads the polymer would not spread out more than necessary for the given conditions, since the length of the necklace is limited. Thus, increasing the number of beads once the sufficient amount has been found would not lead to better results. In equation2.30, we said that the partition function becomes exact using an infinite number of beads. However, our results will be exact, pro-vided that a sufficient number of beads is taken, and we only have to worry about the statistical error. From a physical point of view, a sufficient number of beads is roughly reached when the root mean squared length of the springs is smaller than the relevant length scale of the external potential. This translates into the following relation [26,27]:

P β~^{2}

mσ^{2} (2.39)

with P the number of beads, and σ the length scale of the potential. To a first order, we can say that the number of beads required is inversely propor-tional to the temperature. In practice, the value of P is determined by verifying numerical convergence of an appropriate set of observables (the energy for example).

Up to now, we only presented the path integral formalism and the polymer isomorphism for the special case of one particle in one dimension potential. If we consider N particles in 3 dimensions with interaction between particles, we obtain for the polymer Hamiltonian (following the same procedure as the one dimension case) the following expression:

H_{P} =

P

X

k=1

( _{N}
X

i=1

1 2mi

p^{(k)2}_{i} +1

2m_{i}ω_{P}^{2}(r^{(k+1)}_{i} −r^{(k)}_{i} )^{2}+V(r^{(k)}_{1} ,· · ·,r^{(k)}_{N} )
)

(2.40)

with the conditionr^{(P}_{i} ^{+1)}=r^{(1)}_{i} . An important point to notice is that the potential
V(r^{(k)}_{1} ,· · ·,r^{(k)}_{N} ) only acts between beads of the same index k. The consequence is
that all beads with the same index interact with each other but do not interact with
the others. This behaviour is illustrated in Figure 2.5 for the case of two quantum
particles.

1

2

2

3 4 3

4 1

5

5 6 6

7

7

1

Figure 2.5: Interaction pattern between two quantum particles represented by a ring polymer with P = 7.

This classical isomorphism is the basis of what we call the Path Integral Molecu-lar Dynamics (PIMD) (or Path Integral Monte Carlo: PIMC) method. For example

in PIMD, we let the polymer previously described evolving with the following New-tonian equations (in a one dimensional case but the generalization is staigthforward):

˙

x_{k}= ∂H_{P}

∂pk

˙

p_{k}=−∂H_{P}

∂xk

(2.41)

˙
xk= p_{k}

m p˙k=−mP^{2}

β^{2}~^{2}(2xk−xk+1−xk−1)− ∂V

∂xk

(2.42)

wherem is not necessarly the physical mass.

If the equations 2.41 are coupled to a thermostat (Nosé-Hoover for example), then the dynamics will sample the canonical distribution. Therefore, these classical tra-jectories are used as a sampling device to explore the configuration space of the ring polymer and calculate the exact thermodynamic and structural properties of the system in the limit of a sufficiently large number of beads.

2.1.3 Time-independent equilibrium properties in the canonical