The Virial Identity: Pairwise Forces → Divergence of Stress

The Goal

Show that the internal forces (pairwise interactions) can be written as a divergence of a tensor:

where is the configurational (virial) momentum-flux tensor:

This is the non-obvious bridge from particles to continuum stress.

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Why This Matters

In Newton’s second law for particle :

When we sum over all particles and convert to fields, the external forces become (body force per volume). But what about the internal forces? If we naively write , it’s a sum of point forces at each particle location—not a divergence of anything.

The miracle: Newton’s third law () and clever algebra turn this into . This is how internal forces become stress tensors in the continuum description.

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The Derivation

Step 1: Symmetrize the Pair Sum Using Action–Reaction

Start with:

Rewrite as a sum over unordered pairs with :

(We’re summing each pair twice, so we divide by 2.)

Use (Newton’s third law):

TL;DR

Each pair force appears twice in the full double sum. We can collect them into a symmetric form: one force acting at particle ‘s location and its reaction at particle . This is where Newton’s third law becomes crucial.

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Step 2: The Key Identity — Difference of Deltas as a Divergence

Here’s the non-obvious step. For any pair with :

This is the core identity of this note. Let’s prove it.

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Proof of the Key Identity

Define a “path” from particle to particle :

At : (particle ). At : (particle ).

Define:

Take the derivative w.r.t. :

(Using the chain rule: the only -dependence is in the argument of the delta.)

Integrate from to :

Substitute the expression for :

Convert to a divergence. Using the vector identity (when is constant):

Therefore:

TL;DR

This is the fundamental theorem of calculus applied to deltas along a line segment. We integrate from one particle to the other and use the divergence identity to convert the integrand. Out pops a divergence—which is exactly what we need.

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Step 3: Apply to Pair Forces

Substitute the identity into the symmetrized sum:

Factor out the divergence:

Define the virial tensor:

Result:

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Take-home message

Newton’s third law says . So we can pair up forces: is a force at minus its reaction at . Now, the difference of two deltas can be written as a divergence along the bond connecting them—that’s the calculus step. When we take a divergence, we ‘smear’ the pair force uniformly along the bond, and the divergence recovers the point forces at the ends. This is how pair forces become components of a stress tensor.

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Physical Interpretation: Bonds Carry Stress

Imagine a microscopic ‘bond’ connecting particles and :

The virial tensor represents this:

• Each bond contributes a dyadic product (position times force).
• The integral “smears” this contribution uniformly along the bond.
• When we take a divergence, we recover the traction (force per unit area) on surfaces within the fluid.

Example: Central forces. If (force along the line connecting the particles), then is a symmetric dyad, and the virial tensor contributes to an isotropic stress (like pressure).

Non-central forces (with components perpendicular to the bond) contribute to anisotropic stresses, including shear.

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Component Form

In index notation, the pair force contribution to the momentum balance is:

where

This is the configurational part of the stress tensor. Together with the kinetic part ( from Dyadic-conversion), it forms the total microscopic stress .

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Symmetry of the Stress Tensor

Theorem: If all forces are central (i.e., ), then the Cauchy stress tensor is symmetric: .

Proof sketch: For central forces, is symmetric (the outer product of two parallel vectors). The kinetic part is automatically symmetric (it’s ). So is symmetric, and hence is symmetric.

Why it matters: Symmetric stress is a consequence of angular momentum conservation. If the stress weren’t symmetric, rotational torques wouldn’t be balanced.

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Summary: The Bridge from Particles to Continuum

Level	Quantity	Equation
Particles	Force on particle
Fields (microscopic)	Momentum density
		where
Fields (coarse-grained)	Momentum per volume
		where

The virial identity is the mathematical machinery that takes us from discrete pair forces to a continuous stress field.

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Derivation Summary

1. Symmetrize the pair force sum using .
2. Use the key identity: (follows from FTC).
3. Recognize as a divergence: .

Each step is mathematically rigorous and physically motivated. No approximations until we ensemble-average.

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