Boltzmann Entropy Calculator

S = kB ln(W), where kB is the Boltzmann constant (1.380649e-23 J/K) and W is the number of accessible microstates consistent with a system's macroscopic state. It connects the microscopic (individual particle arrangements) to the macroscopic (measurable entropy), and is inscribed on Ludwig Boltzmann's tombstone in Vienna.

A macrostate is a description in terms of bulk, measurable quantities (total energy, number of particles in a given condition, volume). A microstate is one specific, detailed arrangement of every individual particle that is consistent with that macrostate. Many different microstates usually correspond to the same macrostate, and W counts exactly how many.

It provides the statistical, microscopic foundation for the second law of thermodynamics: since a system overwhelmingly tends to be found in the macrostate with the largest number of microstates (largest W), entropy overwhelmingly tends to increase, not because of some external rule but purely because more-probable arrangements vastly outnumber less-probable ones.

Direct mode takes the microstate count W itself as input and computes S = kB ln(W) in one step, useful when you already know W from some other calculation. Combinatorial mode instead takes N (total particles) and n (number in one of two states) and computes W = N!/(n!(N-n)!), the binomial coefficient, before applying the same formula.

JavaScript's floating-point numbers overflow well before 170!, so computing N! directly is not possible for realistically sized systems. Stirling's approximation for the log-factorial, ln(k!) is approximately k*ln(k) - k + 0.5*ln(2*pi*k), avoids ever forming the enormous factorial itself and remains extremely accurate, especially for N above roughly 20 to 30, which covers essentially every case this calculator is meant for.

The binomial coefficient N!/(n!(N-n)!), the count of distinct arrangements, is largest when n = N/2 (equal population of both states), since that split has the most ways to choose which particles are in which state. Any split away from equal population has fewer equivalent arrangements, hence lower W and lower entropy, this is exactly the curve shown on this page's chart.

Because the Boltzmann constant kB = 1.380649e-23 J/K is itself extremely small. Even when W is astronomically large (say 10^23, on the order of Avogadro's number), ln(W) is only around 53, so S = kB*ln(W) still comes out around 10^-22 J/K, a small number by everyday standards but a perfectly normal entropy value at the atomic scale.

It becomes progressively less precise as N gets very small, since the approximation is derived for large k, but this calculator uses the same Stirling formula throughout its intended range rather than adding a separate exact-factorial code path for small N. For N above roughly 20 to 30 the approximation is excellent, which covers the realistic use cases for this tool.

Joules per Kelvin (J/K), the standard SI unit of thermodynamic entropy, matching the units of the Boltzmann constant itself. Because kB is so small, expect values in Direct mode and Combinatorial mode alike to display in scientific notation, typically somewhere around 10^-22 to 10^-23 J/K for realistic microstate counts.

W represents a count of microstates, so it can never be less than 1 (a system always has at least one way to be arranged). Since ln(1) = 0, the minimum possible entropy for this formula is exactly 0 (a perfectly ordered system with only one accessible microstate), entropy from this formula can never come out negative for any valid W of 1 or more.

The second law states that an isolated system's entropy never decreases over time. Statistically, this follows directly from S = kB ln(W): a system left to evolve overwhelmingly tends toward macrostates with more accessible microstates (higher W, higher S) simply because there are vastly more of them to end up in, not because of any additional physical law beyond probability itself.