{
  "cells": [
    {
      "cell_type": "markdown",
      "id": "markdown-000",
      "metadata": {},
      "source": [
        "# Molecular dynamics simulations\n",
        "\n",
        "[Read the lesson](https://tirta.co/a-byte-of-physics/molecular-dynamics-simulations). This notebook includes the lesson text and its Python examples in reading order. Run the setup cell, then run the remaining cells from top to bottom. In Colab, use File → Save a copy in Drive to keep your edits. You can run and edit this view without changing the course source. Examples use illustrative data unless stated otherwise."
      ]
    },
    {
      "cell_type": "code",
      "id": "code-001",
      "metadata": {},
      "source": [
        "import importlib.metadata\n",
        "import subprocess\n",
        "import sys\n",
        "\n",
        "requirements = {\"numpy\": (2, 0), \"sympy\": (1, 13), \"matplotlib\": (3, 8)}\n",
        "to_install = []\n",
        "for package, minimum in requirements.items():\n",
        "    try:\n",
        "        installed = importlib.metadata.version(package)\n",
        "        version = tuple(int(part) for part in installed.split(\".\")[:2])\n",
        "    except importlib.metadata.PackageNotFoundError:\n",
        "        version = (0, 0)\n",
        "    if version < minimum:\n",
        "        to_install.append(f\"{package}>={minimum[0]}.{minimum[1]}\")\n",
        "if to_install:\n",
        "    subprocess.check_call([sys.executable, \"-m\", \"pip\", \"install\", *to_install])\n",
        "\n",
        "import numpy as np\n",
        "import sympy as sp\n",
        "import matplotlib\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "print(\"Python\", sys.version.split()[0])\n",
        "print(\"NumPy\", np.__version__)\n",
        "print(\"SymPy\", sp.__version__)\n",
        "print(\"Matplotlib\", matplotlib.__version__)\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "id": "markdown-002",
      "metadata": {},
      "source": [
        "Molecular dynamics advances particle positions and velocities from forces, then applies interaction and boundary rules. Its usefulness comes from explicit state, deterministic update order, and diagnostic ledgers—not from a cloud of animated dots. A molecular-dynamics program chooses a force model and an ensemble/boundary policy; those choices determine what energy and momentum conservation claims are valid.\n",
        "\n",
        "## Molecular dynamics simulations in Python\n",
        "\n",
        "Read the relationship as a short program. Begin with measured or chosen values, keep only the state needed for the next calculation, apply the stated rule, and inspect a result that can be checked independently."
      ]
    },
    {
      "cell_type": "code",
      "id": "code-003",
      "metadata": {},
      "source": [
        "import numpy as np\n",
        "\n",
        "# Two Lennard-Jones particles; reduced units epsilon=sigma=mass=1.\n",
        "positions = np.array([[-0.6, 0.0], [0.6, 0.0]])\n",
        "velocities = np.array([[0.0, 0.1], [0.0, -0.1]])\n",
        "\n",
        "\n",
        "def forces(positions):\n",
        "    offset = positions[1] - positions[0]\n",
        "    radius = np.linalg.norm(offset)\n",
        "    if radius < 0.2:\n",
        "        raise ValueError(\"Separation is too small for this demonstration step.\")\n",
        "    radial_derivative = 24 * (radius**-7 - 2 * radius**-13)\n",
        "    first_force = radial_derivative * offset / radius\n",
        "    return np.array([first_force, -first_force])\n",
        "\n",
        "\n",
        "assert np.allclose(forces(positions).sum(axis=0), 0.0)\n",
        "print(\"Initial forces:\", forces(positions))\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "id": "markdown-004",
      "metadata": {},
      "source": [
        "For a conservative pair model, force comes from potential energy. A common programming contract is to define one potential $U(r)$, derive or separately verify its radial force, then sum without double-counting pairs:\n",
        "\n",
        "\n",
        "$$\n",
        "\\mathbf F_{ij}=-\\nabla_{\\mathbf r_i}U(r_{ij}),\\qquad \\mathbf F_{ji}=-\\mathbf F_{ij},\\qquad E=\\sum_i\\tfrac12m_i\\lVert\\mathbf v_i\\rVert^2+\\sum_{i<j}U(r_{ij})\n",
        "$$\n",
        "\n",
        "\n",
        "The $i < j$ convention is not cosmetic: it prevents counting each pair’s potential twice. Force symmetry is a cheap unit test. If a pair loop gives forces that do not sum to zero, total momentum drift may be caused by implementation order rather than an external force. A cutoff, neighbour list, or periodic minimum-image rule changes the approximation and must be stored with the run.\n",
        "\n",
        "Boundaries define the system. Periodic boundaries exchange neither momentum nor energy with an outside world in the usual ideal model; a reflecting wall transfers momentum to the wall; a thermostat intentionally exchanges energy; an imposed shear or temperature gradient is a driven nonequilibrium system. Do not report “energy is not conserved” without first checking whether the model includes one of those transfers.\n",
        "\n",
        "Time step is a physical-numerical compromise. Close repulsive encounters can make a pair potential stiff, so a visually stable run can still have unacceptable energy drift. Hold initial state and output times fixed, then reduce Δt. Compare total-energy drift, momentum residual, minimum separation, and selected aggregate observables. If a thermostat is active, compare the correct extended ledger rather than demanding closed-system energy conservation.\n",
        "\n",
        "A useful check starts with a prediction: **Changing update order inside an interaction loop can create a fake asymmetry, while a synchronous equal-and-opposite pair pass preserves total internal force.** Write a two-particle force fixture and assert $F1 + F2 = 0$. Run the same initial state with Δt, Δt/2, and Δt/4, recording energy and momentum residuals at identical physical times. Then switch from periodic boundaries to a reflecting wall and identify the wall impulse that belongs in the momentum ledger. Finally enable a thermostat and state why a closed-energy assertion is no longer the right test.\n",
        "\n",
        "The explanation above has deliberate limits. Small steps can be expensive, close collisions can be stiff, and simple pair potentials omit real chemistry and quantum behavior. Finite particle counts, cutoff artifacts, long-range interactions, neighbour-list update errors, rigid constraints, phase changes, electronic structure, and non-equilibrium driving may require richer methods. Agreement of a few conservation diagnostics does not validate a chosen potential against a real material.\n",
        "\n",
        "## A standard pair potential\n",
        "\n",
        "The Lennard–Jones model is a common teaching example:\n",
        "\n",
        "\n",
        "$$\n",
        "U(r)=4\\varepsilon\\left[\\left(\\frac{\\sigma}{r}\\right)^{12}-\\left(\\frac{\\sigma}{r}\\right)^6\\right]\n",
        "$$\n",
        "\n",
        "\n",
        "The parameter $\\varepsilon$\n",
        "sets the well depth and $\\sigma$\n",
        "sets the separation where the potential crosses zero. The radial force magnitude follows\n",
        "from $F_r=-dU/dr$.\n",
        "At very small separation the repulsion becomes extremely steep, demanding a smaller time\n",
        "step. This potential is an approximate effective interaction, not a chemical bond model.\n",
        "\n",
        "Truncating interactions at a cutoff saves work but changes both force and energy.\n",
        "Abruptly cutting the potential makes it discontinuous; merely shifting its value makes\n",
        "energy continuous but does not make force continuous. Record whether the potential,\n",
        "force, or both were shifted and test behavior near the cutoff.\n",
        "\n",
        "## Velocity Verlet update\n",
        "\n",
        "A widely used time-reversible, second-order method is velocity Verlet. For acceleration\n",
        "$\\mathbf a_n$,\n",
        "\n",
        "\n",
        "$$\n",
        "\\mathbf r_{n+1}=\\mathbf r_n+\\mathbf v_n\\Delta t+\\frac12\\mathbf a_n\\Delta t^2\n",
        "$$\n",
        "\n",
        "\n",
        "After forces are recomputed at the new positions,\n",
        "\n",
        "\n",
        "$$\n",
        "\\mathbf v_{n+1}=\\mathbf v_n+\\frac12(\\mathbf a_n+\\mathbf a_{n+1})\\Delta t\n",
        "$$"
      ]
    },
    {
      "cell_type": "code",
      "id": "code-005",
      "metadata": {},
      "source": [
        "dt = 0.001\n",
        "initial_momentum = velocities.sum(axis=0)\n",
        "for _ in range(1000):\n",
        "    old_acceleration = forces(positions)\n",
        "    new_positions = positions + velocities * dt + 0.5 * old_acceleration * dt**2\n",
        "    velocities += 0.5 * (old_acceleration + forces(new_positions)) * dt\n",
        "    positions = new_positions\n",
        "assert np.allclose(velocities.sum(axis=0), initial_momentum)\n",
        "print(\"Final positions:\", positions)\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "id": "markdown-006",
      "metadata": {},
      "source": [
        "Do not update one particle completely and then use its new position while other particles\n",
        "still have old positions. That asynchronous order violates the intended simultaneous\n",
        "force snapshot and can make results depend on storage order.\n",
        "\n",
        "## Periodic boundaries and minimum images\n",
        "\n",
        "Periodic boundaries model an infinite repeated cell. A particle crossing one side is\n",
        "wrapped to the opposite side without changing physical velocity. Pair separation uses the\n",
        "nearest periodic image. For a box length $L$,\n",
        "each Cartesian separation component can be mapped into a centered interval before force\n",
        "evaluation.\n",
        "\n",
        "The usual minimum-image method requires a pair cutoff no larger than half the shortest box\n",
        "length. Otherwise more than one image may lie within the cutoff. Long-range electrostatic\n",
        "forces need specialized summation methods rather than a naive short cutoff.\n",
        "\n",
        "Wrapping makes particle coordinates discontinuous at a boundary. Keep unwrapped\n",
        "coordinates or image counters when measuring diffusion; otherwise a particle traveling\n",
        "steadily across the box appears to jump backward.\n",
        "\n",
        "## Ensembles and thermostats\n",
        "\n",
        "An isolated constant-particle, constant-volume, constant-energy calculation approximates\n",
        "the microcanonical ensemble. A thermostat aims to sample a target-temperature ensemble by\n",
        "exchanging energy with a reservoir. It intentionally breaks conservation of the particle\n",
        "system's energy.\n",
        "\n",
        "Simple velocity rescaling can force the instantaneous kinetic temperature to a target but\n",
        "may distort the velocity distribution and dynamics. More advanced deterministic or\n",
        "stochastic thermostats have their own parameters and sampling properties. Always report\n",
        "thermostat energy transfer and validate the resulting distribution, not only its mean\n",
        "temperature.\n",
        "\n",
        "Equilibration data should normally be separated from production measurements. Starting\n",
        "from a lattice or an arbitrary velocity draw creates transients. Decide equilibration\n",
        "using observable histories and correlation times rather than deleting a convenient fixed\n",
        "number of steps without inspection.\n",
        "\n",
        "## From trajectories to observables\n",
        "\n",
        "Temperature, pressure, radial distribution, and diffusion are derived from trajectories.\n",
        "Their samples are correlated. Estimate uncertainty from block averages or independent\n",
        "runs, not by treating every time step as independent.\n",
        "\n",
        "The radial distribution function measures pair separation statistics relative to an\n",
        "uncorrelated reference. For a uniform ideal gas it approaches one away from finite-size\n",
        "and boundary effects. Peaks in an interacting fluid reveal preferred neighbor distances;\n",
        "they are statistical structure, not fixed bonds.\n",
        "\n",
        "Mean-square displacement can estimate diffusion at long times:\n",
        "\n",
        "\n",
        "$$\n",
        "D=\\lim_{t\\to\\infty}\\frac{\\langle|\\mathbf r(t)-\\mathbf r(0)|^2\\rangle}{2dt}\n",
        "$$\n",
        "\n",
        "\n",
        "Use unwrapped positions and verify a linear long-time regime before fitting. Ballistic\n",
        "short-time motion does not satisfy the diffusive approximation.\n",
        "\n",
        "## Verification sequence\n",
        "\n",
        "Start with two particles and compare the coded force with a numerical derivative of the\n",
        "potential. Require equal and opposite forces. Move both particles by the same vector in a\n",
        "periodic cell and require unchanged energy. Permute their storage order and require the\n",
        "same physical output.\n",
        "\n",
        "For a closed conservative run, compare total momentum and total energy at fixed physical\n",
        "times using $\\Delta t$,\n",
        "$\\Delta t/2$, and\n",
        "$\\Delta t/4$. Velocity\n",
        "Verlet should show second-order trajectory convergence over a fixed interval, although\n",
        "long chaotic trajectories separate and should instead be compared statistically.\n",
        "\n",
        "Test neighbor lists against an all-pairs oracle for a small configuration, especially\n",
        "after particles move near the list skin distance. Track minimum separation and reject\n",
        "non-finite forces before they contaminate the full state. With reflecting walls, include\n",
        "wall impulse in momentum accounting. With a thermostat, test its energy ledger and target\n",
        "distribution rather than demanding particle-energy conservation.\n",
        "\n",
        "Molecular dynamics remains classical unless quantum effects are added through another\n",
        "method. Pair potentials can omit electronic polarization, reactions, many-body forces,\n",
        "and nuclear quantum behavior. Conservation and numerical convergence validate an\n",
        "implementation; comparison with material data validates the physical potential.\n",
        "\n",
        "## A worked Python notebook\n",
        "\n",
        "Inspect the repulsive core and attractive tail of a two-particle Lennard-Jones interaction. The question for this calculation is precise: how does **Lennard-Jones potential**\n",
        "change when we change **pair separation**, while keeping the other choices fixed?\n",
        "The earlier sections explain the physical relationship. Here we turn one part of that\n",
        "relationship into a calculation that you can run, inspect, and change in Google Colab.\n",
        "\n",
        "Reduced units set the length and energy scales to one; a cutoff policy changes the implemented model. Keep that sentence beside the results. A formula can return a number outside\n",
        "its useful range, and a plotting library will usually draw that number without asking\n",
        "whether the physical assumptions still hold. The selected interval is from\n",
        "0.95 to 3 reduced length units. It gives us a specific domain to discuss\n",
        "instead of leaving the graph's range to a software default.\n",
        "\n",
        "Use the notebook link above the lesson, run the setup cell, and then run the cells in\n",
        "order. The code below is also included there. NumPy handles the arrays of numerical\n",
        "values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these\n",
        "libraries or notebook cells are unfamiliar, start with\n",
        "[Your first physics notebook](https://tirta.co/a-byte-of-physics/your-first-physics-notebook).\n",
        "\n",
        "### State the relationship before choosing a graph\n",
        "\n",
        "In this small example, `q` stands for pair separation, expressed in reduced length units. The output is\n",
        "Lennard-Jones potential, expressed in reduced energy units. The equation below uses the numerical values in those\n",
        "units. Any fixed coefficients therefore carry the units needed to make the relation\n",
        "consistent. A value supplied in another unit must be converted before it enters this\n",
        "expression.\n",
        "\n",
        "\n",
        "$$\n",
        "f(q) = - \\frac{4}{q^{6}} + \\frac{4}{q^{12}}\n",
        "$$\n",
        "\n",
        "\n",
        "This is a function of one changing input. It is not a statement that the physical\n",
        "system has only one variable. We deliberately hold the other inputs fixed so that one\n",
        "cause of variation is visible. Changing several inputs at once would make it harder\n",
        "to tell which change produced the output difference. Write down the held-fixed choices\n",
        "before trying a different experiment.\n",
        "\n",
        "At 1.2 reduced length units, the expression gives -0.890965 reduced energy units. This is our\n",
        "reference calculation. It is a prediction of the stated example, not a new measurement.\n",
        "Its displayed digits help us compare implementations. They should not be read as a\n",
        "claim that physical inputs are known to six significant figures."
      ]
    },
    {
      "cell_type": "code",
      "id": "code-007",
      "metadata": {},
      "source": [
        "import numpy as np\n",
        "import sympy as sp\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "q = sp.Symbol(\"q\", real=True)\n",
        "expression = sp.sympify(\"4*(q**-12-q**-6)\", locals={\"q\": q})\n",
        "reference_input = 1.2\n",
        "reference_output = float(expression.subs(q, reference_input))\n",
        "print(\"Input:\", reference_input, \"reduced length units\")\n",
        "print(\"Predicted Lennard-Jones potential:\", reference_output, \"reduced energy units\")\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "id": "markdown-008",
      "metadata": {},
      "source": [
        "`sp.Symbol` declares a mathematical symbol. It does not assign a numerical measurement\n",
        "to that symbol. `subs` then replaces the symbol with our chosen input, and `float`\n",
        "converts the resulting symbolic number into a standard Python numerical value.\n",
        "Keeping these steps separate lets us inspect the equation before numerical substitution.\n",
        "It also prevents a later plot from becoming the only surviving record of the model.\n",
        "\n",
        "### Compare a few cases you can explain\n",
        "\n",
        "Before drawing hundreds of points, calculate three. The lower endpoint gives\n",
        "1.96097 reduced energy units. The midpoint, at 1.975 reduced length units, gives\n",
        "-0.0662639 reduced energy units. The upper endpoint gives -0.00547944 reduced energy units.\n",
        "Read these together with the assumptions: they describe different choices of pair separation\n",
        "within the same example, with the other inputs held fixed.\n",
        "\n",
        "These three values are a useful first check, but they do not establish the shape between\n",
        "them. Two endpoints can be equal even when the curve bends or oscillates in between.\n",
        "Likewise, a large change between endpoints does not tell us whether the relationship is\n",
        "linear. We need the equation and intermediate samples to answer those questions."
      ]
    },
    {
      "cell_type": "code",
      "id": "code-009",
      "metadata": {},
      "source": [
        "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n",
        "inputs = np.array([0.95, 1.975, 3.0])\n",
        "outputs = np.asarray(evaluate(inputs), dtype=float)\n",
        "outputs = np.broadcast_to(outputs, inputs.shape)\n",
        "for input_value, output_value in zip(inputs, outputs):\n",
        "    print(\n",
        "        f\"{input_value:.6g} reduced length units -> \"\n",
        "        f\"{output_value:.6g} reduced energy units\"\n",
        "    )\n",
        "assert np.all(np.isfinite(outputs))\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "id": "markdown-010",
      "metadata": {},
      "source": [
        "`lambdify` turns the symbolic expression into a numerical function. We explicitly select\n",
        "NumPy as its numerical backend so it can accept an array. The same expression now has\n",
        "two useful forms: one for algebra and one for calculating many cases. The assertion\n",
        "checks that the chosen inputs produce finite outputs. It would catch an infinity or an\n",
        "undefined numerical result, but a finite value can still be physically inappropriate.\n",
        "\n",
        "### Make a graph that answers the question\n",
        "\n",
        "The horizontal axis shows pair separation; the vertical axis shows Lennard-Jones potential. Every plotted\n",
        "point is a prediction of the same expression. We mark the reference case separately\n",
        "so that a reader can connect the earlier printed value to its place on the curve.\n",
        "The axis labels include units because a shape without a scale cannot tell us the size\n",
        "of a physical effect."
      ]
    },
    {
      "cell_type": "code",
      "id": "code-011",
      "metadata": {},
      "source": [
        "input_grid = np.linspace(0.95, 3.0, 301)\n",
        "output_grid = np.asarray(evaluate(input_grid), dtype=float)\n",
        "output_grid = np.broadcast_to(output_grid, input_grid.shape)\n",
        "fig, ax = plt.subplots(figsize=(7, 4))\n",
        "ax.plot(input_grid, output_grid, label=\"Stated model\")\n",
        "ax.scatter([reference_input], [reference_output], label=\"Reference case\", zorder=3)\n",
        "ax.set_xlabel(\"Pair separation (reduced length units)\")\n",
        "ax.set_ylabel(\"Lennard-jones potential (reduced energy units)\")\n",
        "ax.legend()\n",
        "ax.grid(alpha=0.25)\n",
        "fig.tight_layout()\n",
        "plt.show()\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "id": "markdown-012",
      "metadata": {},
      "source": [
        "Look for a sign change, a turning point, or a region where the curve becomes steep.\n",
        "Connect each feature to the expression rather than to the colour of the line. If the\n",
        "curve has a minimum or maximum, changing the input in opposite directions may have\n",
        "similar effects on the output near that point. If it is nearly straight over a short\n",
        "interval, a local linear approximation may be useful even when the complete expression\n",
        "is nonlinear.\n",
        "\n",
        "Increasing the number of samples makes the line smoother and may reveal features that\n",
        "a coarse grid missed. It does not improve the physical inputs or enlarge the model's\n",
        "validity range. For quantities that can take only discrete values, use the line as a\n",
        "guide to the relationship and interpret only the allowed input values. A plotting\n",
        "function does not enforce such restrictions for you.\n",
        "\n",
        "### Measure the local sensitivity\n",
        "\n",
        "A derivative asks a more specific question than the whole graph: how quickly does\n",
        "Lennard-Jones potential change near the reference input? Its units are the output unit divided by\n",
        "the input unit. For this calculation, the derivative at 1.2 reduced length units is\n",
        "2.21169 in those ratio units. Locally, the output increases as the\n",
        "input increases. This statement is local; the sign or size of the slope can be different\n",
        "elsewhere on the curve.\n",
        "\n",
        "\n",
        "$$\n",
        "f'(q) = \\frac{24}{q^{7}} - \\frac{48}{q^{13}}\n",
        "$$\n",
        "\n",
        "\n",
        "For a small input change, multiply the derivative by that change to estimate the output\n",
        "change. Then compare that estimate with a fresh evaluation of the full expression.\n",
        "The distinction between these two calculations matters: the derivative gives a tangent\n",
        "approximation, while substitution evaluates the chosen expression at the new point."
      ]
    },
    {
      "cell_type": "code",
      "id": "code-013",
      "metadata": {},
      "source": [
        "slope_expression = sp.diff(expression, q)\n",
        "slope_at_reference = float(slope_expression.subs(q, reference_input))\n",
        "changed_input = 1.21025\n",
        "linear_prediction = reference_output + slope_at_reference * (\n",
        "    changed_input - reference_input\n",
        ")\n",
        "full_prediction = float(evaluate(changed_input))\n",
        "print(\"Symbolic derivative:\", slope_expression)\n",
        "print(\"Tangent prediction:\", linear_prediction, \"reduced energy units\")\n",
        "print(\"Full expression:\", full_prediction, \"reduced energy units\")\n",
        "print(\n",
        "    \"Approximation difference:\",\n",
        "    full_prediction - linear_prediction,\n",
        "    \"reduced energy units\",\n",
        ")\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "id": "markdown-014",
      "metadata": {},
      "source": [
        "Here the changed input is 1.21025 reduced length units. The tangent estimate is\n",
        "-0.868295 reduced energy units, while the full expression gives\n",
        "-0.867848 reduced energy units. If the two values are close, it means the chosen\n",
        "input change is small enough for this local approximation at this point. It does not\n",
        "mean the entire model is linear. If the slope is zero, a quadratic or higher-order term\n",
        "may supply the leading change instead.\n",
        "\n",
        "This is also the beginning of uncertainty propagation. If the input has a small stated\n",
        "standard uncertainty, the magnitude of the slope times that uncertainty estimates one\n",
        "contribution to output uncertainty. Do not automatically use the same rule for a wide\n",
        "interval, a discontinuity, a turning point with zero first derivative, or several\n",
        "correlated uncertain inputs. Those situations need more of the model than one slope.\n",
        "\n",
        "### Check the derivative with numerical differences\n",
        "\n",
        "We can estimate the same slope from values on either side of the reference point.\n",
        "Subtract the lower output from the upper output and divide by twice the input step.\n",
        "Parentheses are essential: the entire output difference must be divided by the entire\n",
        "input interval. Without them, Python can execute a different mathematical expression\n",
        "while still producing a plausible number.\n",
        "\n",
        "The calculation below uses three decreasing step sizes. Each estimate is compared with\n",
        "the symbolic derivative, so the reference is independent of the numerical difference\n",
        "formula. Near an endpoint, it is acceptable here to evaluate a small extension only\n",
        "because the displayed mathematical expression is smooth there; such a check does not\n",
        "assert that extrapolated values describe additional physical cases."
      ]
    },
    {
      "cell_type": "code",
      "id": "code-015",
      "metadata": {},
      "source": [
        "input_span = 2.05\n",
        "for fraction in [1e-2, 1e-3, 1e-4]:\n",
        "    step = input_span * fraction\n",
        "    upper = float(evaluate(reference_input + step))\n",
        "    lower = float(evaluate(reference_input - step))\n",
        "    numerical_slope = (upper - lower) / (2 * step)\n",
        "    difference = numerical_slope - slope_at_reference\n",
        "    print(f\"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}\")\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "id": "markdown-016",
      "metadata": {},
      "source": [
        "For a smooth function, reducing the step often reduces the approximation error at first.\n",
        "Very small steps can eventually make subtraction of nearly equal floating-point values\n",
        "less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare\n",
        "the observed behaviour over a useful range of steps instead of choosing the smallest\n",
        "number the computer can represent.\n",
        "\n",
        "An absolute slope difference is useful when the expected slope is near zero. A relative\n",
        "difference would divide by a very small reference and could look enormous even when the\n",
        "absolute discrepancy is tiny. Away from zero, a relative difference can help compare\n",
        "quantities with very different scales. The tolerance should reflect the question and\n",
        "the numerical method, rather than a universal decimal threshold copied into every test.\n",
        "\n",
        "### Change the example and preserve the reasoning\n",
        "\n",
        "Make one copy of the reference calculation before editing it. Choose a second value of\n",
        "pair separation inside the stated interval, predict whether Lennard-Jones potential will rise or fall,\n",
        "and then evaluate the expression. Place the prediction and the result in a text cell.\n",
        "If they disagree, identify whether the mistake was in the sign, the unit conversion,\n",
        "the algebra, or the assumption about how the curve behaves.\n",
        "\n",
        "Next choose a shorter plotting interval around the reference point. Compare the local\n",
        "curve with its tangent approximation. Then widen the interval again. The difference\n",
        "shows why a useful local estimate need not describe the whole range. Keep the axis\n",
        "labels and numerical scale visible when comparing the plots; automatic axis limits can\n",
        "make two different effects appear equally large.\n",
        "\n",
        "Finally, return to the physical limit stated at the start: Reduced units set the length and energy scales to one; a cutoff policy changes the implemented model. Describe one change\n",
        "to the setup that would violate that limit. State what new input, equation, or observation\n",
        "would be needed before trusting an answer in the changed situation. This last step\n",
        "connects the Python experiment to the physics instead of treating successful execution\n",
        "as the only criterion for a useful result.\n",
        "\n",
        "Before saving, restart the runtime and run all cells in order. A result that depends on\n",
        "an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep\n",
        "unrounded values for later calculations, round only the reported output, and label any\n",
        "new data as measured, assumed, or generated. The graph and printed values should remain\n",
        "traceable to the same expression and the same choices of inputs.\n",
        "\n",
        "### Recover the output change by integrating the slope\n",
        "\n",
        "There is another way to check that the expression and its derivative describe the same\n",
        "relationship. Add up the small output changes predicted by the slope across the chosen\n",
        "input interval. In the continuous limit, this integral equals the output at the upper\n",
        "endpoint minus the output at the lower endpoint. The result has units of reduced energy units:\n",
        "the input unit in the integration cancels the input unit in the denominator of the slope.\n",
        "\n",
        "This statement is about the mathematics of this particular relationship. It does not\n",
        "assign a new physical meaning to an arbitrary area under a graph. For example, an area\n",
        "under force versus distance represents work, while an area under a different pair of\n",
        "quantities may represent something else or have no standard name. Always carry the\n",
        "axis units through the calculation before naming an integral."
      ]
    },
    {
      "cell_type": "code",
      "id": "code-017",
      "metadata": {},
      "source": [
        "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n",
        "endpoint_change = float(evaluate(3.0) - evaluate(0.95))\n",
        "for points in [101, 201, 401]:\n",
        "    grid = np.linspace(0.95, 3.0, points)\n",
        "    slopes = np.asarray(slope_function(grid), dtype=float)\n",
        "    slopes = np.broadcast_to(slopes, grid.shape)\n",
        "    integrated_change = np.trapezoid(slopes, grid)\n",
        "    print(\n",
        "        points,\n",
        "        \"integrated change:\",\n",
        "        integrated_change,\n",
        "        \"endpoint change:\",\n",
        "        endpoint_change,\n",
        "    )\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "id": "markdown-018",
      "metadata": {},
      "source": [
        "The trapezoidal rule joins neighbouring slope samples with straight segments and adds\n",
        "their signed areas. A negative contribution remains negative; replacing every slope\n",
        "by its magnitude would calculate a different quantity. Increasing the sample count\n",
        "provides a practical refinement check. If the values do not settle toward the endpoint\n",
        "difference, first inspect the interval, singularities, and array ordering.\n",
        "\n",
        "Do not conclude that two agreeing calculations prove the model is physically complete.\n",
        "Both use the same selected relationship, so an omitted effect would be omitted from\n",
        "both. Their agreement is evidence that the numerical tools implement that relationship\n",
        "consistently. Comparing with an independent experiment or a more appropriate physical\n",
        "model is a separate test.\n",
        "\n",
        "Keep this comparison in the notebook when you change the expression. It is particularly\n",
        "useful after adding a coefficient, changing a sign convention, or converting an input\n",
        "unit. A code cell that produces a smooth graph can conceal these mistakes; a second\n",
        "calculation of the same output change gives you another place to notice them.\n",
        "\n",
        "## Sources\n",
        "- M. P. Allen and D. J. Tildesley, *Computer Simulation of Liquids*.\n",
        "- D. Frenkel and B. Smit, *Understanding Molecular Simulation*.\n",
        "- [OpenStax, *University Physics Volume 2*, Chapter 2: The Kinetic Theory of Gases](https://openstax.org/books/university-physics-volume-2/pages/2-introduction).\n",
        "- [MIT 8.01SC, Chapter 29: Kinetic Theory of Gases](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter29/).\n",
        "- [Benjamin Crowell, *Light and Matter*, kinetic theory and microscopic models](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n",
        "- [The Physics Hypertextbook, “Kinetic-Molecular Theory”](https://physics.info/kmt/)."
      ]
    }
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