An Upper Bound on the Probability of a Macroscopic Food Object (a “Kraft Single”) Arising as a Boltzmann Fluctuation
We bound the probability that thermal fluctuations in the observable universe have ever produced, for any nonzero duration, a specific macroscopic object: a ~20 g processed cheese slice (“Kraft Single”). Modeling the event as a large negative entropy fluctuation out of near-equilibrium matter, we combine (i) a conservative lower bound on the entropy decrease required for coarse localization of ~10²⁴ atoms into a fixed 10⁻⁵ m³ volume with (ii) an extremely generous upper bound on the total number of statistically independent spacetime trials (Planck-scale 4-cells across the observable universe and its age). The resulting expectation ( \mathbb E[N] ) and total probability (P) obey
[ \mathbb E[N];\lesssim;P;\lesssim;10^{-4.34\times 10^{24}}, ]
i.e., effectively zero for any physical purpose. Tighter (more realistic) modeling only decreases (P).
Question. What is the probability that a specific, recognizable macroscopic object—a 20 g sliced processed cheese piece with ordinary density and composition—appears spontaneously from thermal fluctuations (“Boltzmann fluctuation”) anywhere in the observable universe over cosmic time?
Approach. Treat the event as an entropy-reducing fluctuation with probability per statistically independent trial
[ p_{\text{trial}}\sim \exp!\left(-\frac{\Delta S}{k}\right), ]
where ( \Delta S>0 ) is the entropy decrease required to assemble the object. Then
[ P ;\le; N_{\text{trials}}; p_{\text{trial}}. ]
Chosen to maximize the chance:
- Thermal bath: Uniform, favorable temperature (T=300) K everywhere.
- Feedstock availability: Required molecules always present.
- Recognition criterion: Only mass, bulk volume, and coarse shape matter.
- Independence granularity: Every Planck 4-cell counts as an independent trial.
For (N\sim 10^{24}) atoms localized into volume (V_{\text{slice}}\sim 10^{-5}\ \text{m}^3) from (V_{\text{avail}}=1\ \text{m}^3):
[ \frac{\Delta S}{k} \approx N \ln!\frac{V_{\text{avail}}}{V_{\text{slice}}} \sim 10^{24}\times 11.5 \sim 10^{25}. ]
Chemical specificity, bonding, folding, and microstructure are ignored (all would increase (\Delta S)).
Conservative bound:
[
\Delta S/k ;\gtrsim; 10^{25}.
]
[ p_{\text{trial}};\lesssim;\exp(-10^{25}). ]
In base-10: [ p_{\text{trial}};\lesssim;10^{-4.34\times 10^{24}}. ]
- Observable universe volume: (3.6\times 10^{80}\ \text{m}^3).
- Planck volume: (4.2\times 10^{-105}\ \text{m}^3).
- Spatial cells: (\sim 10^{185}).
- Cosmic time / Planck time: (\sim 10^{61}).
Total 4-cells:
[
N_{\text{trials}} \lesssim 10^{246}.
]
[ P \approx N_{\text{trials}}p_{\text{trial}} \lesssim 10^{246 - 4.34\times 10^{24}} \approx 10^{-4.34\times 10^{24}}. ]
- Planck 4-cell independence vastly overcounts trials.
- (V_{\text{avail}}=1\ \text{m}^3) chosen artificially small.
- Ignored chemical/microstructural entropy terms.
- Optimistic uniform temperature assumption.
All corrections only make (P) smaller.
Even with unrealistically favorable assumptions, the probability that a 20 g Kraft Single has arisen as a Boltzmann fluctuation anywhere in the observable universe is bounded by
[ P ;\lesssim; 10^{-4.34\times 10^{24}}. ]
Formally nonzero, but indistinguishable from zero for any physical purpose.
- Atoms: (10^{24}).
- Object volume: (10^{-5}\ \text{m}^3).
- Available volume: (1\ \text{m}^3).
- Entropy penalty: (\Delta S/k \sim 10^{25}).
- Per-trial probability: (10^{-4.34\times 10^{24}}).
- Universe trials: (10^{246}).