T1 Relaxation from First Principles: Stochastic BPP Simulation

Static Field & Motion

Field B₀ 1.5 T

Tumbling correlation time τ_c 2.5 ns

Pulse Sequence

Live Readouts

near T₁ minimum

T₁ (quadrature)

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ω₀τ_c

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FT quadrature err

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MC vs analytic RMS

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How this differs from the 3D version

The 3D simulator renders the BPP formula directly onto a rotating dipole lattice. This 2D companion instead derives the same T₁ from three independent numerical experiments, run live and cross-checked against each other:

Panel 1 — stochastic proxy. An Ornstein-Uhlenbeck random walk b(s), in reduced time s = t/τ_c, stands in for the fluctuating local dipolar field produced by molecular tumbling (rotational diffusion).
Panel 2 — Wiener-Khinchin check. The empirical autocorrelation of the simulated b(s) trace is measured directly from the trajectory and overlaid on the analytic e^-Δs it is supposed to have — proof the random-number generator actually reproduces the assumed correlation structure, not just that the formula was typed in.
Panel 3 — numerical Fourier transform. J(ω) = τ_c/(1+ω²τ_c²) is re-derived by Simpson-quadrature integration of ∫exp(-x/τ_c)cos(ωx)dx rather than evaluated from the closed form, at both ω₀ and 2ω₀, then combined into 1/T₁ = K·[J(ω₀)+4J(2ω₀)] exactly as BPP theory prescribes.
Panel 4 — microscopic Monte Carlo. A population of independent two-level spins each fire a single stochastic relaxation event at rate 1/T₁ (Poisson process); their ensemble average is not forced to be exponential — it emerges from many random individual flips, and is plotted against the analytic M_z(t)=M₀(1−2e^−t/T1) it should match.

J(ω) = ∫0∞ e^(-x/τc)·cos(ωx) dx   (Simpson quadrature)
1/T1 = K·[J(ω0) + 4·J(2ω0)]
per-spin: P(flip -1→+1 in dt) = dt/T1

Move the sliders and every panel updates together: the T₁-vs-τ_c curve still traces the same V-shaped BPP minimum at ω₀τ_c≈1 as the 3D sim, but every number behind it here comes from a distinct computational path — quadrature and Monte Carlo — rather than the closed-form formula alone.

Local field b(s) — OU trace Empirical autocorrelation Analytic / quadrature Spin: relaxed (+1) Spin: inverted (−1)