Lecture 7 - Isothermal Simulation & Boltzmann Distribution

Overview

Nathan Seifert connects molecular-dynamics simulations to the statistical meaning of temperature, first comparing argon and water with radial distribution functions (RDFs). Using a thermostat to bring systems toward thermal equilibrium, he shows how water’s hydrogen bonds and molecular motions produce structure and energy fluctuations absent from the dilute argon example, then derives the Boltzmann distribution: populations of energy states vary as exp[−βΔE], where β = 1/(kBT).

Key takeaways

An RDF turns molecular-dynamics trajectories into a structural measurement: preferred neighbor distances produce peaks, while an ideal-gas-like distribution approaches a uniform long-range baseline.
The 25-atom argon box and 20-water example show how intermolecular forces change structure: hydrogen bonding creates a prominent water RDF peak near 3 angstroms even under comparable thermodynamic conditions.
Thermal equilibrium is best identified by a stable average energy, not a perfectly flat instantaneous energy trace, because thermostats exchange heat as molecular energy fluctuates.
Water’s translational, rotational, and vibrational motions create more energy-transfer pathways than argon’s motion, producing more complex short-term energy fluctuations.
The Boltzmann distribution assigns energy states exponentially different populations: Nⱼ/N₀ = exp[−(Eⱼ−E₀)/(kBT)], so higher-energy states become less probable at a fixed temperature.

Chapters

0:00 Biochemistry Simulations: Proteins, Drugs, and Periodic Boxes

Molecular dynamics can test whether a drug binds to or inhibits a protein in water.
Typical setup specifies temperature, pressure, and simulation-box volume; periodic boundary conditions make the finite box repeat as an approximation of a bulk sample.
Pressure, volume, temperature, and concentration help determine how many water and solute molecules to include.

7:39 Force Fields Approximate Molecular Interactions

A simulation can represent water as charged atoms, or model a protein at a coarser resolution, depending on the question.
Force fields encode effective interactions—including hydrogen-bond attractions and repulsions—because full quantum-mechanical calculations are too expensive for large systems.
Biochemical simulations may combine separate force fields for water, a protein, and a drug molecule.

11:11 RDFs Measure Local Molecular Structure

A radial distribution function (RDF) summarizes how often neighboring atoms occur at different distances from a reference atom.
The calculation gathers interatomic distances across atoms and simulation snapshots, then plots distance against the observed frequency or normalized probability.
Peaks in an RDF can reveal preferred neighbor distances and molecular shells.

16:22 The Ideal-Gas Benchmark for Neighbor Distributions

For a dilute ideal gas, particles are treated as noninteracting and spatially random, so the long-range RDF should approach a uniform baseline.
Short-distance exclusions can occur because particles cannot occupy the same location; preferred shells would indicate additional structure.
Seifert uses this expectation to compare an argon simulation with water.

20:53 Machine-Learned Potentials and Simulation Controls

The argon demonstration uses a 20-angstrom-sided box containing 25 atoms and a machine-learned interatomic potential.
The browser-based computational chemistry tool sets the number of steps, time interval, target temperature, and thermostat.
A thermostat exchanges energy with the system to maintain a target temperature, while time steps must be short enough to resolve molecular motion.

26:30 Argon RDFs and Finite-Sample Fluctuations

The simulated argon RDF shows a near-neighbor feature followed by weaker distance-dependent structure.
With only 25 atoms in a dilute box, the RDF is jagged because the simulation supplies limited observations.
Longer simulations and larger samples would improve statistics; the long-distance behavior approaches the expected uniform-gas baseline.

33:19 Thermostat Feedback Makes Energy Fluctuate

A thermally equilibrated system does not require a perfectly constant instantaneous energy; the relevant signal is a stable long-term average.
The thermostat repeatedly adds or removes energy as the system fluctuates, producing oscillations in the energy trace.
Seifert compares the feedback lag to a hot plate overshooting and undershooting its temperature target, a behavior he calls hysteresis.

35:53 Water’s Hydrogen Bonds Create Neighbor Shells

Unlike argon atoms, water molecules attract through hydrogen bonds, which can associate nearby molecules.
A water molecule can donate and accept hydrogen bonds, with up to four neighboring waters in the idealized coordination picture.
These interactions predict strong RDF peaks and gaps corresponding to successive hydration shells.

39:58 Water’s Rotations and Vibrations Complicate Energy Traces

The water simulation’s energy fluctuates more rapidly than argon’s as translational, rotational, and vibrational motions exchange energy.
Water’s bending vibrations and molecular rotations provide more ways to distribute energy than the motion of a spherical argon atom.
The long-term energy trend still approaches a stable average even when instantaneous behavior looks chaotic.

42:26 Water RDFs Show Structure That Argon Lacks

The 20-water example produces a prominent near-neighbor RDF peak at roughly 3 angstroms, consistent with hydrogen-bond association.
The small sample does not show a clear second shell, but its sharp first peak contrasts with argon’s more uniform long-range behavior.
Comparing systems at the same temperature, volume, and density shows that molecular interactions—not thermodynamic conditions alone—shape structure.

45:53 Heating Water Weakens Its Hydrogen-Bond Structure

Seifert raises the water simulation target to 1,000 K and observes molecules becoming more dispersed as the system heats.
The resulting RDF retains some short-range association but has a more uniform long-distance distribution.
The comparison illustrates that gas behavior can become more ideal at higher temperature, while lower-temperature argon can also develop structure.

49:07 Temperature Makes Molecular Averages Reproducible

Setting a system to a target temperature gives a reproducible reference for comparing its average properties across runs.
Seifert shifts from simulation examples to the statistical question of how molecular energies are distributed after equilibration.
The derivation begins with isolated compartments assigned different, discrete energy levels.

59:58 Thermal Contact Redistributes Energy Across Compartments

The model uses a closed outer container with compartments that exchange heat but do not exchange particles.
Initially, each compartment contains particles assigned a particular discrete energy; a water bath at temperature T then allows heat to flow among compartments.
After equilibration, the compartments share a common temperature, and their energy populations are no longer uniformly weighted.

1:03:27 Energy-Ratio Consistency Leads to an Exponential

Seifert defines ratios of compartment populations at different energies and assumes each ratio depends on the energy difference.
Consistency across adjacent and multi-step energy transitions requires the ratio function to turn sums of energy differences into products of ratios.
That functional property identifies an exponential dependence; the lower probability of higher-energy states supplies the negative sign.

1:11:39 Boltzmann Distribution Connects Energy, Temperature, and kB

The resulting population ratio is Nⱼ/N₀ = exp[−β(Eⱼ−E₀)], with β = 1/(kBT) and E₀ used as a reference energy.
Boltzmann’s constant is approximately 1.380649 × 10⁻²³ J/K per particle; multiplying it by Avogadro’s number gives the molar gas constant R.
On a molar energy basis, the same weighting can be written exp(−ΔE/RT); Seifert previews using it for calculations and revisiting water’s non-ideal behavior.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Nathan Seifert.