Boltzmann Entropy Calculator
Enter number of microstates W to compute thermodynamic entropy S=k·ln(W), information entropy bits=log₂(W) and negentropy. W=1024 → 10 bits.
Input Data
Results
At a glance:The Boltzmann entropy formula (Ludwig Boltzmann, 1877; engraved on his Vienna tombstone): S=k_B·ln(W). k_B=1.380649×10⁻²³ J/K (Boltzmann constant, exactly defined in 2019), W is the number of microstates corresponding to a macrostate (thermodynamic probability). Physical meaning: entropy measures the number of ways to realize a macrostate — the larger W, the more disordered the system and the more information missing to an outside observer, so S∝ln(W). Second law: an isolated system tends toward the macrostate with the largest W (most probable), hence entropy increases. Information-theory link: Shannon entropy H=−Σpᵢ·log₂pᵢ; for the equiprobable case H=log₂(W) bits; thus S=k_B·ln(2)·H — thermodynamic entropy differs from information entropy only by the constant k_B·ln2. Negentropy (Schrödinger, 1944): the information content of an ordered system = H_max−H; 'life feeds on negative entropy.' History: Boltzmann derived it in 1877 from the kinetic theory of gases, resolving Maxwell's demon paradox; Gibbs generalized it to S=−kΣpᵢ ln pᵢ; Shannon established information entropy in 1948; Landauer's principle (1961): erasing 1 bit releases k_B·T·ln2 of heat. Classic example: W=2 (two-state) → S=k·ln2≈9.57×10⁻²⁴ J/K, H=1 bit; W=1024=2¹⁰ → 10 bits; W=8.7×10⁶⁷ (shuffled deck) → 226 bits; 1 mol ideal gas W~10²³ → S~205 J/(mol·K). Applications: (1) the second law and arrow of time; (2) data compression and coding; (3) black-hole entropy S=k·A/(4l_P²) (Bekenstein-Hawking); (4) biological negentropy and metabolism; (5) computational thermodynamics (Landauer heat).
Formula
Boltzmann entropy: S = k·ln(W)
Information entropy: bits = log₂(W) = S / (k·ln2)
Negentropy: negentropy = −bits
Gibbs entropy: S = −k·Σ pᵢ·ln(pᵢ)
Landauer heat: Q = k·T·ln2 (per bit)
$$S = k_B \ln W, \quad \text{bits} = \frac{S}{k_B \ln 2} = \log_2 W, \quad Q_{\text{Landauer}} = k_B T \ln 2$$How to Use
- Enter the number of microstates W (≥1).
- The tool computes S (J/K), information entropy bits and negentropy.
Microstates and Entropy Examples
| Scenario | W | bits | S (J/K) |
|---|---|---|---|
| 1 coin | 2 | 1 | 9.57e-24 |
| 1 die | 6 | 2.58 | 2.47e-23 |
| 10 coins | 1024 | 10 | 9.57e-23 |
| 52-card deck | 8.7e67 | 226 | 2.16e-21 |
| 1 mol gas | 1e6e23 | 46.5 | 205 |
S=k·ln(W). bits=log₂(W). Daily W is enormous so entropy is usually in J/K or cal/K. 1 mol gas entropy ~205 J/(mol·K).
Case Studies
Maxwell's Demon and Information Entropy
Maxwell (1867) imagined a demon separating fast and slow molecules, seemingly violating the second law by decreasing entropy. The Boltzmann-Szilard resolution: the demon must acquire information about molecular speeds, and erasing that information inevitably releases k·T·ln2 of heat (Landauer).
Thus the total entropy (system + demon's memory) never decreases, and the second law holds. This reveals the equivalence of information and thermodynamics.
Modern application: Landauer's principle is the lower bound of computation energy; as Moore's law approaches it, chip heating becomes a fundamental limit.
Biological Negentropy and Metabolism
Schrödinger (1944, 'What is Life?') proposed: life feeds on negative entropy — taking in low entropy (ordered food) and expelling high entropy (waste heat, CO₂).
A human takes in ~10⁶ J of free energy per day, expelling waste entropy ΔS≈−10⁵ J/K (equivalent to ~10²³ bits of information).
Photosynthesis converts low-entropy sunlight into ordered glucose. DNA stores genetic information (human genome ~6e9 bits). Life is a locally entropy-decreasing system; the universe's total entropy still increases.
FAQ
Why does S=k·ln(W) use a logarithm?
For two independent systems the total number of microstates is W=W₁·W₂, but entropy should add (extensivity): S=S₁+S₂. The logarithm satisfies ln(W₁·W₂)=lnW₁+lnW₂, so S∝lnW. The logarithmic choice gives entropy its additive (extensive) property. Boltzmann's choice aligns with the information content in probability theory.
How do bits convert to J/K?
1 bit of entropy = k_B·ln2 ≈ 9.57×10⁻²⁴ J/K. So S(bits)=S/(k_B·ln2)=log₂(W). 1 bit of information equals 9.57e-24 J/K of entropy. Landauer's principle: erasing 1 bit at room temperature releases k_B·T·ln2 ≈ 2.9e-21 J (at 300 K) — tiny, but accumulating over 1e9 operations per second becomes significant.
Why does the second law always hold?
Microscopically: an isolated system evolves from a low-W state to a high-W state with overwhelmingly larger probability (W_high/W_low is astronomically large). Macroscopically this is entropy increase. It is statistical certainty, not absolute: very small systems can fluctuate (e.g. Brownian motion). But for macroscopic systems (1e23 molecules) the reverse probability is 1e-10^23, practically impossible. This gives rise to the arrow of time.
Why is black-hole entropy proportional to area?
Bekenstein (1973) proved black-hole entropy S=k·A/(4l_P²), where A is the event-horizon area, in units of Planck area l_P². Thus black holes store information on their surface (max density 1 bit/l_P²). This 'area entropy' differs from ordinary volume entropy and hints at the holographic principle (the universe's information is stored on a lower-dimensional boundary). The black-hole information paradox — where information goes after evaporation — remains unsolved.
What does negentropy mean?
A term coined by Schrödinger, it is the deviation from maximum entropy: the negentropy of an ordered system = H_max − H. Living organisms are internally highly ordered (low entropy), having negentropy relative to their disordered environment. Through metabolism they take in negentropy (low-entropy food, light) to maintain structure and expel waste entropy (heat). It is not 'negative entropy' but 'entropy lower than the maximum', equivalent to information content.
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References
Content reviewed by the Calculatorism editorial team. Results are for reference only; please refer to the relevant authorities for the official figures.