Materials Genomics
Unit 5: Monte Carlo Sampling & Continuum Mechanics

Prof. Dr. Philipp Pelz

FAU Erlangen-Nürnberg

Where We Stand

Recap of Units 1–4

Unit 1: postulates of QM, hydrogen atom, Born interpretation
Unit 2: multi-electron atoms, Born–Oppenheimer, Slater determinants, LCAO
Unit 3: Hartree–Fock, post-HF (MP, CC), density functional theory
Unit 4: thermodynamics, statistical mechanics, classical interatomic potentials, molecular dynamics
We can now produce finite-temperature trajectories of \(10^4\)–\(10^7\) atoms — but only if we trust ergodicity to deliver equilibrium averages

What Unit 5 Adds — The Pivot

MD samples the canonical distribution by integrating Newton — slow when barriers \(\gg k_B T\)
A complementary sampler: Monte Carlo (MC) moves through configuration space without dynamics
For length scales above the atomic, we move to a continuum description — fields, PDEs, balance laws
We learn the two workhorses for solving those PDEs numerically: finite differences and a brief look at finite elements / volumes
Together with Units 1–4, this completes the simulation toolkit that feeds the ML half of the course

Lecture Roadmap

Part I — Monte Carlo sampling

The sampling problem and importance sampling
Markov chains, detailed balance, Metropolis
Ergodicity, equilibration, move design
Ensembles, kinetic MC, MC vs MD

Part II — Continuum & discretization

Balance laws and constitutive relations
Finite Difference Method (FDM)
Weak forms, Finite Element Method (FEM)
Finite Volumes (FV) and outlook

A Multiscale View

Length	Time	Method
0.1–1 Å	fs	QM / DFT (Units 1–3)
Å–nm	fs–ps	classical MD (Unit 4)
Å–μm	μs–s	Monte Carlo (today, Part I)
nm–m	ms–years	Continuum / FEM (today, Part II)

The “right” method tracks the smallest feature you must resolve
ML in materials genomics plays at every scale — surrogates for DFT, MLIPs for MD, neural PDE solvers for FEM
Today’s content is what those surrogates ultimately reproduce

Part I — Monte Carlo Sampling

The Sampling Problem

Equilibrium thermodynamics computes ensemble averages

\[\langle A\rangle = \int A(\mathbf{r}^N,\mathbf{p}^N)\,p_{\text{Boltz}}(\mathbf{r}^N,\mathbf{p}^N)\,d\mathbf{r}^N d\mathbf{p}^N\]

For \(N\sim 10^4\) atoms, the integral is over \(\sim 6\times 10^4\) dimensions
Closed-form solutions exist only for trivially decoupled systems
We need a sampling strategy instead of a quadrature strategy
Goal: draw configurations \(\{(\mathbf{r}^N_m,\mathbf{p}^N_m)\}_{m=1}^M\) such that \(\langle A\rangle\approx \tfrac{1}{M}\sum_m A_m\)

Why Brute Force Fails

For a regular grid with \(n\) points per dimension:

\[\text{cost} = n^{3N}\]

\(N=10\) atoms, \(n=10\): \(10^{30}\) evaluations — infeasible
This is the curse of dimensionality
Even sparse grids and quasi-Monte-Carlo lattices cannot beat the exponential scaling for typical materials problems
We need to sample adaptively, putting compute where the integrand is large

Why Uniform Sampling Fails

Uniform random sampling: pick \(\mathbf{r}^N\) at random with probability \(1/V^N\).

Almost every uniform draw places atoms on top of each other \(\Rightarrow\) astronomical \(V\), vanishing Boltzmann weight \(e^{-V/k_BT}\approx 0\)
The Boltzmann mass concentrates in an exponentially small subset of phase space
Estimator variance grows exponentially with \(N\)
Effectively no signal — every sample contributes essentially zero

We must bias the sampler toward high-probability regions: importance sampling.

Importance Sampling — The Idea

If we can draw \(\mathbf{x}_m \sim p(\mathbf{x})\) directly, then

\[\langle A\rangle = \int A(\mathbf{x})\,p(\mathbf{x})\,d\mathbf{x}\;\approx\;\frac{1}{M}\sum_{m=1}^{M} A(\mathbf{x}_m)\]

The estimator collapses to a simple average when samples follow \(p\)
Variance is now \(\mathrm{Var}(A)/M\) instead of an exponentially bad worst case
The target distribution here is the Boltzmann distribution \[p_{\text{Boltz}}(\mathbf{x}) = \frac{1}{Z}\,e^{-\mathcal{H}(\mathbf{x})/k_B T}\]
The catch: we cannot draw from \(p_{\text{Boltz}}\) directly because \(Z\) is intractable

The Partition Function Problem

The partition function is itself an integral over all of phase space \[Z = \int e^{-\mathcal{H}(\mathbf{x})/k_B T}\,d\mathbf{x}\]
Computing \(Z\) has exactly the same dimensionality problem as the original average
Key observation: ratios of probabilities do not depend on \(Z\) \[\frac{p_{\text{Boltz}}(i)}{p_{\text{Boltz}}(j)} = \exp\!\left[-\frac{E_i - E_j}{k_BT}\right]\]
A sampler that uses only ratios can target \(p_{\text{Boltz}}\) without ever evaluating \(Z\)
Markov-chain Monte Carlo (MCMC) is exactly such a sampler

Markov Chains — Setup

A Markov chain is a sequence of random states \(X_0,X_1,X_2,\dots\) with

\[P(X_{t+1}=x'\mid X_0,\dots,X_t) = P(X_{t+1}=x'\mid X_t) \equiv T(x_t\to x')\]

Memoryless: the next state depends only on the current state
\(T\) is the transition kernel — a probability over \(x'\) for each \(x\)
A distribution \(\pi\) is stationary if pushing it through \(T\) leaves it unchanged
If the chain is ergodic (irreducible + aperiodic), it converges to \(\pi\) from any starting state

Stationary Distributions

\(\pi\) is stationary if and only if

\[\pi(x') = \sum_x \pi(x)\,T(x\to x')\qquad\text{(global balance)}\]

A stronger, easier-to-design condition is detailed balance:

\[\pi(x)\,T(x\to x') = \pi(x')\,T(x'\to x)\]

Sum over \(x\) on both sides: detailed balance \(\Rightarrow\) global balance. The converse is not true — detailed balance is sufficient, not necessary.

Detailed Balance

Detailed balance demands that probability flux between every pair of states cancels:

\[\pi(i)\,P(i\to j) = \pi(j)\,P(j\to i)\]

Imposes a local condition rather than a global sum
Almost trivial to verify for a proposed acceptance rule
Sufficient (with ergodicity) to guarantee convergence to \(\pi\)
The standard design recipe for equilibrium MCMC algorithms

Proposal × Acceptance

Decompose the transition kernel:

\[P(i\to j) = P_q(i\to j)\,P_a(i\to j)\]

\(P_q\): proposal — how we suggest a trial move (random displacement, swap, …)
\(P_a\): acceptance — whether we keep the trial move
Inserting into detailed balance and dividing yields \[\frac{P_a(i\to j)}{P_a(j\to i)} = \frac{\pi(j)\,P_q(j\to i)}{\pi(i)\,P_q(i\to j)}\]
For symmetric proposals, \(P_q(i\to j) = P_q(j\to i)\), the proposal cancels

Deriving the Metropolis Criterion

For symmetric proposals, the acceptance ratio must satisfy

\[\frac{P_a(i\to j)}{P_a(j\to i)} = \exp\!\left[-\frac{E_j - E_i}{k_B T}\right]\]

The simplest rule satisfying this is the Metropolis acceptance:

\[\boxed{\;P_a(i\to j) = \min\!\left[1,\;\exp\!\left(-\frac{\Delta E}{k_B T}\right)\right]\;}\]

Down-hill moves (\(\Delta E\leq 0\)): always accepted
Up-hill moves (\(\Delta E>0\)): accepted with Boltzmann probability

The Metropolis Algorithm

Metropolis, Rosenbluth, Teller (1953).

Initialise the system in some state (random, lattice, snapshot, …)
Propose a symmetric trial move \(i\to j\) (e.g. displace one atom by \(\delta\in[-\Delta,\Delta]^3\))
Compute the energy difference \(\Delta E = E_j - E_i\)
Draw \(u\sim\mathcal{U}(0,1)\). Accept if \(u<\min[1,e^{-\Delta E/k_BT}]\)
If rejected, stay in \(i\) and count it again
After equilibration, average observables along the chain

Why Metropolis Works

With symmetric proposals, the Metropolis rule satisfies detailed balance for \(\pi=p_{\text{Boltz}}\):

\[\frac{P(i\to j)}{P(j\to i)} = \frac{P_a(i\to j)}{P_a(j\to i)} = e^{-\Delta E/k_BT} = \frac{\pi(j)}{\pi(i)}\]

The chain therefore converges to the canonical distribution
Accepting unfavourable moves lets it escape local minima
The rule never needs \(Z\) — only differences of energies
“Reject = stay = count again” preserves the proper invariant measure

Ergodicity

Detailed balance fixes the invariant distribution, but ergodicity is needed for convergence:

The chain must be able to reach every state with nonzero \(\pi\) in finite time (irreducibility)
It must not get stuck in a cyclic pattern (aperiodicity)
Proposal moves must be rich enough to cross all relevant barriers
Conservative moves \(\Rightarrow\) the chain may explore only a basin and give wrong averages
In practice: combine multiple move types (local + global) and check independence of the answer to the starting state

MC Workflow — Equilibration & Production

Equilibration / burn-in: discard the first \(N_{\text{eq}}\) samples — they reflect the initial state, not \(\pi\)
Production: record observables for \(M\) samples
Block averaging or autocorrelation analysis: estimator variance is set by independent samples, not total samples
Effective sample size \(M_\text{eff} = M / (1 + 2\tau)\) with autocorrelation time \(\tau\)

Run multiple independent chains from different seeds
Compare within-chain and between-chain variance (Gelman–Rubin \(\hat R\))
Diagnose poor mixing before trusting the average
Hint for the exercise: \(\tau\) is often much larger than you think

Tuning Step Size

Acceptance rate \(\alpha\) as a function of step size \(\Delta\):

Tiny \(\Delta\): \(\alpha\to 1\) but each step barely moves — slow exploration
Huge \(\Delta\): almost every proposal rejected, \(\alpha\to 0\)
Sweet spot: 20–50% acceptance for atomistic systems (theoretical optimum 23% for Gaussian targets)
Adapt \(\Delta\) during equilibration; freeze it for production
Auto-tuning during production breaks detailed balance — separate the phases

Move Catalogue for Materials

Atom displacement: random \(\delta\) on one atom — workhorse for fluids and amorphous systems
Species swap: exchange the labels of two atoms — fast equilibration of order/disorder
Volume / cell move: rescale box and coordinates — required for NPT
Particle insertion / removal: required for grand-canonical (\(\mu\)VT) sampling
Configurational-bias moves: regrow polymer chains segment by segment
Cluster moves: flip whole connected clusters (Swendsen–Wang) — Ising-type problems
Hybrid MC: short MD trajectory used as a proposal, Metropolis accept/reject

Ensembles in MC

Ensemble	Fixed	Extra move type
NVT (canonical)	\(N,V,T\)	displacement
NPT	\(N,P,T\)	+ volume move
\(\mu\)VT (grand canonical)	\(\mu,V,T\)	+ insertion / removal
Gibbs (multi-box)	\(N_\text{tot},V_\text{tot},T\)	+ particle transfer

Same Metropolis acceptance, generalised by the appropriate Boltzmann weight
\(\mu\)VT and Gibbs are unique to MC — MD cannot create/destroy atoms
Phase coexistence (e.g. liquid–vapour) is naturally obtained in Gibbs MC
Mirror the experimental boundary conditions

MC vs MD — Side by Side

MD strengths

True dynamics: diffusion, viscosity, spectra
Non-equilibrium relaxation
Forces already available from MLIPs
Familiar Newtonian intuition

MC strengths

No forces required, only energies
“Unphysical” moves: species swap, volume change, particle exchange
Grand-canonical sampling
Often faster equilibrium sampling for dense / stiff systems
Trivially parallel over independent chains

In practice they are complementary: MD for kinetics, MC for equilibrium and rare events.

Metropolis–Hastings & Beyond

Asymmetric proposal \(\Rightarrow\) acceptance ratio carries the proposal correction:

\[P_a(i\to j) = \min\!\left[1,\;\frac{\pi(j)\,P_q(j\to i)}{\pi(i)\,P_q(i\to j)}\right]\]

Foundation for biased moves, parallel tempering, replica exchange
Kinetic Monte Carlo (kMC): choose events with rates \(r_{i\to j}\), advance the clock by \(\Delta t = -\ln u / \sum r\)
kMC samples kinetics of rare events (diffusion, defect migration) when a rate table is available
Modern variants: Hamiltonian / hybrid MC, NUTS, normalising-flow proposals — all share the same backbone

Part II — Continuum Mechanics

Why Continuum?

Below ~1 nm: atomistic discreteness matters \(\Rightarrow\) MD / MC / DFT
Above ~1 μm: \(10^{15}\) atoms is hopeless. Treat matter as a continuous medium
Fields \(\rho(\mathbf{x},t)\), \(T(\mathbf{x},t)\), \(\mathbf{u}(\mathbf{x},t)\), \(\sigma(\mathbf{x},t)\) live at every point
Atomic information enters only through material laws (constitutive relations)
This is the regime of structural mechanics, heat transport, fluid flow, electromagnetism
Same mathematical machinery serves every conserved quantity: mass, momentum, energy, charge, …

1D Balance Setup

Take a small interval \([x, x+dx]\), with flux \(q(x)\) flowing through the boundary and a source/sink density \(s(x)\):

Influx from source: \(s(x)\,dx\)
Net flux through boundaries: \(q(x+dx)-q(x)\)
If the two balance, total inside is constant

\[s\,dx = q(x+dx) - q(x)\] \[\Rightarrow s = \frac{q(x+dx)-q(x)}{dx}\]

Taking \(dx\to 0\) gives the differential form.

Stationary Continuity Equation

Limit of the balance:

\[\boxed{\;s = \frac{\partial q}{\partial x}\;}\]

Net divergence of flux equals the source
“What flows out must be produced inside”
Valid for any conserved quantity at steady state
Independent of the physical meaning of \(q\) — particles, heat, momentum, money

Transient Continuity Equation

If influx and out-flux don’t balance, the amount inside changes in time. Defining number density \(\rho_{\text{nmbr}} = N/dx^{\text{dim}}\) and taking \(dx,dt\to 0\):

\[\frac{\partial \rho_{\text{nmbr}}}{\partial t} = -\frac{\partial q}{\partial x} + s\]

The first term moves material in/out
The second adds material from sources
This is the transient continuity equation in 1D
Generalises directly to higher dimensions

Multi-Dimensional Form

In \(d\) dimensions the flux becomes a vector \(\mathbf{q}\in\mathbb{R}^d\) and the divergence replaces \(\partial/\partial x\):

\[\boxed{\;\frac{\partial \rho}{\partial t} = -\nabla\cdot\mathbf{q} + s\;}\]

where

\[\nabla\cdot\mathbf{q} = \sum_{i=1}^{d}\frac{\partial q_i}{\partial x_i}\]

This is the universal form of conservation. To close the system we need constitutive laws that specify \(\mathbf{q}\) in terms of the field.

Constitutive Laws

Constitutive laws connect flux to gradient of the field:

Fick’s law (diffusion): \[\mathbf{q} = -D\,\nabla c\]
Fourier’s law (heat): \[\mathbf{q} = -k\,\nabla T\]
Darcy’s law (porous flow): \[\mathbf{q} = -k_{\text{hyd}}\,\nabla p\]

All have the same form: flux opposes gradient
Coefficients (\(D\), \(k\), \(k_{\text{hyd}}\)) carry the material identity
Combining with continuity yields the diffusion / heat / Darcy PDEs
e.g. \(\partial_t c = D\nabla^2 c + s\)

Isotropy vs Anisotropy

Scalar coefficient \(\Rightarrow\) isotropic material — same response in all directions
Many real materials are not: layered composites, fibre-reinforced polymers, single-crystal metals
General form uses a material tensor \(\mathbf{A}\): \[\mathbf{q} = -\mathbf{A}\,\nabla\varphi\]
\(\mathbf{A}\) has up to \(d^2\) independent entries (symmetry usually reduces this)
The principal axes of \(\mathbf{A}\) are the directions of fastest / slowest transport
ML potentials and ML constitutive models routinely predict \(\mathbf{A}\) from microstructure

Part III — Discretization

Why Discretize?

Analytic solutions exist only for highly symmetric, often academic geometries
Real engineering problems involve complex shapes, multiphase media, nonlinearities
We replace the continuous PDE by an algebraic system on a finite grid
Two big families: finite differences (point values on a regular grid) and finite elements (basis functions on a mesh)
Both share a foundation: Taylor expansion of the field

Taylor Expansion

For a smooth scalar field \(\varphi\):

\[\varphi(x_0+\delta) = \sum_{j=0}^{\infty}\frac{1}{j!}\frac{\partial^j\varphi(x_0)}{\partial x^j}\,\delta^j\]

Truncating at order \(J\) leaves an error \(\mathcal{O}(\delta^{J+1})\)
We will obtain derivative approximations by rearranging truncated expansions
The choice of which neighbours to use determines accuracy and stability
All FDM stencils, including the gold-standard Crank–Nicolson and Runge–Kutta schemes, derive from this template

Forward & Backward Differences

Truncating at first order:

Forward:

\[\frac{\partial\varphi(x_0)}{\partial x} \approx \frac{\varphi(x_0+\delta) - \varphi(x_0)}{\delta}\]

Backward:

\[\frac{\partial\varphi(x_0)}{\partial x} \approx \frac{\varphi(x_0) - \varphi(x_0-\delta)}{\delta}\]

Both have leading error \(\mathcal{O}(\delta)\) — only first-order accurate
One-sided; useful at boundaries
Cheap and stable for advection-dominated problems with upwinding

Central Difference

Subtract the backward from the forward expansion: the second-order terms cancel.

\[\boxed{\;\frac{\partial\varphi(x_0)}{\partial x} \approx \frac{\varphi(x_0+\delta) - \varphi(x_0-\delta)}{2\delta}\;}\]

Leading error \(\mathcal{O}(\delta^2)\) — one order better than one-sided
Symmetric: no preferred direction
Requires neighbour information on both sides \(\Rightarrow\) needs special treatment at boundaries
Standard choice for diffusive PDEs

Second-Derivative Stencil

Add forward and backward expansions to order 2: first-order terms cancel.

\[\frac{\partial^2\varphi(x_0)}{\partial x^2} \approx \frac{\varphi(x_0+\delta) - 2\varphi(x_0) + \varphi(x_0-\delta)}{\delta^2}\]

Error \(\mathcal{O}(\delta^2)\) (odd-order terms cancel)
Three-point symmetric stencil
Generalises to higher derivatives and higher orders
This is the discrete Laplacian — appears in heat, Poisson, Schrödinger, Helmholtz, …

Worked Example: 1D Laplace

Solve \(\partial^2\varphi/\partial x^2 = 0\) on \(x\in[0,4]\) with \(\varphi(0)=0\), \(\varphi(4)=1\).

Three grid points, \(\delta=2\):

\[x_1=0,\; x_2=2,\; x_3=4\]

Boundary values: \(\varphi_1=0\), \(\varphi_3=1\).

Central stencil at interior node:

\[\frac{\varphi_3 - 2\varphi_2 + \varphi_1}{\delta^2} = 0\;\;\Rightarrow\;\;\varphi_2 = \tfrac{1}{2}\]

The discrete problem became a single algebraic equation
More interior nodes \(\Rightarrow\) a tridiagonal linear system
Larger meshes: sparse linear solvers (e.g. conjugate gradient)
Same structure carries to 2D / 3D Poisson — the workhorse of FDM

Boundary Conditions

Three families dominate practice:

Dirichlet: value of \(\varphi\) is prescribed on \(\partial\Omega\) — fixed temperature, fixed concentration
Neumann: normal derivative \(\partial\varphi/\partial n\) is prescribed — insulated wall, fixed flux
Robin (mixed): linear combination \(\alpha\varphi + \beta\,\partial\varphi/\partial n = \gamma\) — convective heat transfer
One-sided differences are used at the edge of the domain
Bad boundary handling is the single most common bug in undergraduate FDM code

Time Stepping & CFL

For transient problems, march in time with finite differences as well:

Forward (explicit) Euler: \(\varphi^{n+1} = \varphi^n + \Delta t\,F(\varphi^n)\) — cheap, but conditionally stable
Backward (implicit) Euler: \(\varphi^{n+1} = \varphi^n + \Delta t\,F(\varphi^{n+1})\) — needs a linear solve, unconditionally stable
Crank–Nicolson: average of the two — second-order accurate
For diffusion with explicit Euler: stable only if \(\Delta t \leq \delta^2/(2D)\)
For advection: CFL condition \(\Delta t\leq \delta/|v|\) — information cannot move faster than the grid

Toward FEM: Weak Form

Multiply the PDE by an arbitrary test function \(w\) and integrate over the domain:

\[\int_\Omega \frac{\partial}{\partial x}\!\left(A(x)\,\frac{\partial\varphi}{\partial x}\right)\,w\,dx = \int_\Omega q\,w\,dx\]

Integrate the left-hand side by parts to lower the order of differentiation on \(\varphi\):

\[-\int_\Omega A\,\frac{\partial\varphi}{\partial x}\,\frac{\partial w}{\partial x}\,dx + \text{boundary terms} = \int_\Omega q\,w\,dx\]

The PDE is now stated as an integral identity for all admissible \(w\). Discretise both \(\varphi\) and \(w\) on a finite basis \(\Rightarrow\) a linear system.

Finite Element Method

Approximate \(\varphi(x)\approx\sum_i \varphi_i\,N_i(x)\) with local shape functions \(N_i\) (typically piecewise polynomials on a mesh).

Mesh handles complex 2D/3D geometry
Local support \(\Rightarrow\) sparse stiffness matrix
Higher-order \(N_i\) trade memory for accuracy (h- vs p-refinement)
Natural treatment of Neumann boundaries through the boundary term in the weak form
Industry standard: ANSYS, Abaqus, COMSOL, Code_Aster, FEniCS, deal.II

Strength: complex geometries, mixed BCs, multiphysics couplings
Weakness: mesh generation can dominate the workflow
Cousin: spectral methods (global shape functions, exponential accuracy on smooth domains)

Finite Volumes Variant

Set the test function \(w\equiv 1\) on each control volume and apply the divergence theorem:

\[\int_{V_k}\nabla\cdot\mathbf{q}\,dV = \oint_{\partial V_k}\mathbf{q}\cdot\hat{\mathbf{n}}\,dA\]

The PDE becomes a local flux balance between neighbouring cells
Conservative by construction — total mass / energy is preserved to round-off
Method of choice for CFD (OpenFOAM, ANSYS Fluent), reservoir engineering, shock-capturing schemes
Conceptually closest to the original “balance law” derivation we started with

Closing

Where ML Plugs In

Surrogate force fields (MACE, M3GNet, CHGNet, …) deliver \(E,\,\mathbf{F}\) for MD/MC at ~DFT accuracy
Neural samplers (Boltzmann generators, normalising flows) propose MC moves that vault over barriers — Noé et al. 2019
Neural PDE solvers (PINNs, Fourier neural operators, graph neural PDE solvers) learn the discretised continuum operator
Operator learning (FNO, DeepONet) generalises across geometries — promising for FEM-scale problems
Inverse design treats the continuum solver as a differentiable layer in a generative loop
Every technique in the rest of this lecture series sits on top of the simulation tools we just introduced

Key Takeaways

Importance sampling: integrate by drawing from a target distribution, not on a grid
Markov-chain MC + detailed balance: design a chain whose invariant distribution is \(p_{\text{Boltz}}\) — Metropolis is the canonical example
Ergodicity is non-negotiable: move design and diagnostics determine whether averages are physical
Continuum mechanics replaces atoms with fields. Conservation gives a continuity equation; constitutive laws close the system
FDM turns derivatives into Taylor stencils; FEM/FV start from a weak form / flux balance to handle real geometries
These three sampling paradigms (DFT, MD, MC, FEM) span the length-scale ladder. ML in materials genomics learns surrogates for each rung

Outlook — Unit 6

Unit 6: how do we encode a local atomic environment so an ML model can read it?
Classical descriptors (Magpie, matminer, RDF) and the local-environment ladder (ACSF, SOAP, ACE)
Those descriptors power universal ML force fields (MACE-MP-0, M3GNet, CHGNet) that feed straight back into the MD/MC machinery built today
Unit 7 then takes the conceptual leap to graph-based representations (CGCNN, MEGNet, SchNet, ALIGNN) and message passing on periodic crystal graphs

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Materials GenomicsUnit 5: Monte Carlo Sampling & Continuum Mechanics