Gibbs Entropy Calculator

Compute Gibbs entropy S = -kB sum(pi ln pi) for any probability distribution over up to 5 discrete states, the general formula that Boltzmann's S=kB ln(W) is a special case of.

🔀 Gibbs Entropy Calculator

Leave unused states at 0. All 5 values must sum to 1.

Gibbs entropy (S)
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S / kB
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States with p>0
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Step-by-step working

🔀 What is the Gibbs Entropy Calculator?

This Gibbs entropy calculator computes entropy from the Gibbs entropy formula, S=-kB sum(pi ln pi), for an arbitrary probability distribution over up to 5 discrete microstates. Enter each state's probability, and it returns the entropy in joules per Kelvin, along with the dimensionless sum in "nats" and a step-by-step breakdown of every term.

Unlike Boltzmann's famous S=kB ln(W), which only works when every one of the W microstates is equally likely, the Gibbs formula works for any probability distribution, equal or unequal. In fact, Boltzmann's formula is a special case of Gibbs's: substitute pi=1/W into the Gibbs sum for all W equally-likely states, and it reduces exactly to S=kB ln(W). That makes the Gibbs formula the more fundamental, general statement of statistical entropy, with Boltzmann's version as the equal-probability special case.

A common point of confusion is what happens when a state's probability is exactly 0. Mathematically, ln(0) is undefined, so a direct calculation of 0 × ln(0) in most software produces NaN (not a number). The correct convention, used throughout statistical mechanics and information theory, is that the limit of p·ln(p) as p approaches 0 is exactly 0, so a zero-probability state simply contributes nothing to the sum. This calculator applies that convention explicitly.

This calculator is useful for statistical mechanics and information theory students exploring entropy beyond the equally-likely-microstates special case, and for anyone who wants to see the maximum entropy principle (why entropy is highest when probability is spread evenly) demonstrated numerically and visually.

📐 Formula

S  =  −kB·∑i pi·ln(pi)
kB = 1.380649×10−23 J/K (Boltzmann constant, exact SI-defined value)
pi = probability of microstate i, with ∑i pi = 1
Convention: if pi = 0, the term pi·ln(pi) is taken to be exactly 0 (the mathematical limit as p→0)
Example: p = [0.25, 0.25, 0.25, 0.25]: S = kB·ln(4) ≈ 1.91399×10−23 J/K, exactly matching Boltzmann's formula for 4 equally-likely states.

📖 How to Use This Calculator

Steps

1
Enter up to 5 probabilities - Leave any unused states at 0. All entered probabilities must sum to 1.
2
Click Calculate - The calculator validates that the probabilities sum to 1 before computing.
3
Read the Gibbs entropy - See S=-kB sum(pi ln pi) in joules per Kelvin, with the working shown step by step.

💡 Example Calculations

Example 1 - Uniform 4-state distribution

1
p = [0.25, 0.25, 0.25, 0.25, 0], a system with 4 equally-likely states
2
Each term: 0.25×ln(0.25) = −0.346574, and there are 4 such terms
3
∑ pi·ln(pi) = −1.386294
4
S = −kB×(−1.386294) = 1.91399×10−23 J/K, exactly equal to kB·ln(4), matching the Boltzmann-formula result for 4 equally-likely microstates
S = 1.91399 × 10^-23 J/K
Try this example →

Example 2 - Non-uniform 4-state distribution

1
p = [0.5, 0.3, 0.15, 0.05, 0], the same 4 states but with unequal probabilities
2
Terms: 0.5×ln(0.5)=−0.346574, 0.3×ln(0.3)=−0.361192, 0.15×ln(0.15)=−0.284568, 0.05×ln(0.05)=−0.149787
3
∑ pi·ln(pi) = −1.142120
4
S = −kB×(−1.142120) = 1.57687×10−23 J/K, noticeably lower than Example 1's uniform-distribution result for the same 4 states, exactly what the maximum entropy principle predicts: entropy is highest when probability is spread as evenly as possible
S = 1.57687 × 10^-23 J/K
Try this example →

Example 3 - Certain outcome (zero entropy)

1
p = [1, 0, 0, 0, 0], the outcome is completely certain to be state 1
2
1×ln(1) = 0, and every other term is 0 by the p=0 convention
3
S = −kB×0 = 0 J/K, the minimum possible entropy, a perfectly ordered, zero-uncertainty distribution
S = 0 J/K
Try this example →

❓ Frequently Asked Questions

What is Gibbs entropy?+
Gibbs entropy is S = -kB sum(pi ln pi), the general statistical-mechanics formula for entropy given any probability distribution pi over a system's accessible microstates. It works whether the microstates are equally likely or not, making it more general than Boltzmann's S=kB ln(W).
How does Gibbs entropy relate to Boltzmann entropy?+
Boltzmann's S=kB ln(W) assumes all W microstates are equally likely, each with probability p=1/W. Substituting pi=1/W into the Gibbs formula for all W states gives exactly S=kB ln(W), so Boltzmann's formula is the special equal-probability case of the more general Gibbs formula.
Why does entropy peak for the uniform distribution?+
This is the maximum entropy principle: for a fixed number of possible outcomes, spreading probability as evenly as possible across them maximizes uncertainty, and hence entropy. The chart on this page illustrates this for a simple 2-outcome system, peaking exactly at p1=0.5.
What happens when a probability is exactly 0?+
By mathematical convention, a state with pi=0 contributes exactly 0 to the sum, since the limit of p*ln(p) as p approaches 0 is 0, even though ln(0) itself is undefined. This calculator applies that convention explicitly rather than letting a naive multiplication produce an invalid result.
What units does Gibbs entropy come out in?+
Joules per Kelvin (J/K), the same SI unit as Boltzmann entropy, since both use the same Boltzmann constant kB=1.380649e-23 J/K. The dimensionless sum -sum(pi ln pi), sometimes called the entropy in 'nats', is also shown on this page for reference.
Why must the probabilities sum to 1?+
Because pi represents the probability of the system being found in state i, and the system must be in exactly one of its accessible states. If your inputs do not sum to 1 (within a small tolerance), this calculator shows an error rather than a misleading answer.
What does it mean if Gibbs entropy comes out to exactly 0?+
S=0 only when one state has probability 1 and every other state has probability 0, meaning the outcome is completely certain. This is the minimum possible entropy, a perfectly ordered, zero-uncertainty distribution.
Can this calculator handle fewer than 5 states?+
Yes, leave any unused probability inputs at 0. A 0 probability contributes nothing to the sum (by the p ln p convention above), so a 3-state or 2-state distribution works correctly by simply leaving the remaining inputs at 0, as long as the entered probabilities still sum to 1.
Is the illustrative chart a plot of my actual input probabilities?+
No, the chart shows a simplified 2-outcome system (p1 and p2=1-p1) swept across its full range, since the calculator's actual 5-variable input cannot be meaningfully plotted along a single axis. It is included to visually demonstrate the maximum entropy principle, not as a live plot of your specific inputs above.
How is Gibbs entropy different from Shannon entropy in information theory?+
Mathematically they are the same formula, H=-sum(pi log pi), just applied in different contexts and often with a different logarithm base and no Boltzmann constant. Claude Shannon's 1948 information entropy was directly inspired by, and shares the exact mathematical form of, Gibbs's earlier statistical-mechanical entropy.
Why is Gibbs entropy considered more fundamental than Boltzmann entropy?+
Because it applies to any probability distribution over microstates, equal or unequal, while Boltzmann's formula only applies to the special equal-probability case. Every result from Boltzmann's formula can be derived from Gibbs's, but not the other way around, which is why Gibbs entropy is the more general and foundational statement.
Tags: Gibbs Entropy Calculator Gibbs Entropy Formula Statistical Entropy Maximum Entropy Principle Calculator Formula Free Online

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What is Gibbs entropy?

Gibbs entropy is S = -kB sum(pi ln pi), the general statistical-mechanics formula for entropy given any probability distribution pi over a system's accessible microstates. It works whether the microstates are equally likely or not, making it more general than Boltzmann's S=kB ln(W).

How does Gibbs entropy relate to Boltzmann entropy?

Boltzmann's S=kB ln(W) assumes all W microstates are equally likely, each with probability p=1/W. Substituting pi=1/W into the Gibbs formula for all W states gives exactly S=kB ln(W), so Boltzmann's formula is the special equal-probability case of the more general Gibbs formula.

Why does entropy peak for the uniform distribution?

This is the maximum entropy principle: for a fixed number of possible outcomes, spreading probability as evenly as possible across them maximizes uncertainty, and hence entropy. The chart on this page illustrates this for a simple 2-outcome system, peaking exactly at p1=0.5.

What happens when a probability is exactly 0?

By mathematical convention, a state with pi=0 contributes exactly 0 to the sum, since the limit of p*ln(p) as p approaches 0 is 0, even though ln(0) itself is undefined. This calculator applies that convention explicitly rather than letting a naive multiplication produce an invalid result.

What units does Gibbs entropy come out in?

Joules per Kelvin (J/K), the same SI unit as Boltzmann entropy, since both use the same Boltzmann constant kB=1.380649e-23 J/K. The dimensionless sum -sum(pi ln pi), sometimes called the entropy in 'nats', is also shown on this page for reference.

Why must the probabilities sum to 1?

Because pi represents the probability of the system being found in state i, and the system must be in exactly one of its accessible states. If your inputs do not sum to 1 (within a small tolerance), this calculator shows an error rather than a misleading answer.

What does it mean if Gibbs entropy comes out to exactly 0?

S=0 only when one state has probability 1 and every other state has probability 0, meaning the outcome is completely certain. This is the minimum possible entropy, a perfectly ordered, zero-uncertainty distribution.

Can this calculator handle fewer than 5 states?

Yes, leave any unused probability inputs at 0. A 0 probability contributes nothing to the sum (by the p ln p convention above), so a 3-state or 2-state distribution works correctly by simply leaving the remaining inputs at 0, as long as the entered probabilities still sum to 1.

Is the illustrative chart a plot of my actual input probabilities?

No, the chart shows a simplified 2-outcome system (p1 and p2=1-p1) swept across its full range, since the calculator's actual 5-variable input cannot be meaningfully plotted along a single axis. It is included to visually demonstrate the maximum entropy principle, not as a live plot of your specific inputs above.

How is Gibbs entropy different from Shannon entropy in information theory?

Mathematically they are the same formula, H=-sum(pi log pi), just applied in different contexts and often with a different logarithm base and no Boltzmann constant. Claude Shannon's 1948 information entropy was directly inspired by, and shares the exact mathematical form of, Gibbs's earlier statistical-mechanical entropy.

Why is Gibbs entropy considered more fundamental than Boltzmann entropy?

Because it applies to any probability distribution over microstates, equal or unequal, while Boltzmann's formula only applies to the special equal-probability case. Every result from Boltzmann's formula can be derived from Gibbs's, but not the other way around, which is why Gibbs entropy is the more general and foundational statement.

📌 Quick Tips

💡Gibbs entropy is the general formula for statistical entropy, it works for any probability distribution, not just one where every microstate is equally likely.
💡When every state has equal probability p=1/N, the Gibbs formula reduces exactly to Boltzmann's S=kB ln(W), a special case of the more general Gibbs result.
💡By convention, a state with probability exactly 0 contributes exactly 0 to the sum, since the mathematical limit of p ln(p) as p approaches 0 is 0.
💡Entropy is maximized when probabilities are spread as evenly as possible across the available states, and minimized (S=0) when the outcome is certain, this is the maximum entropy principle.
💡The FAQ section below answers the most common questions about Gibbs entropy.