Mathematics · Probability
Markov Detailed-Balance Flow Residual Calculator
Calculate detailed-balance residual from forward stationary transition flow and reverse stationary transition flow.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Use c=a−b with forward stationary transition flow=0.12 and reverse stationary transition flow=0.115.
- detailed-balance residual=0.0049999999999999906.
Understand Markov Detailed-Balance Flow Residual
One idea, three depths
Choose how deeply to explain Markov Detailed-Balance Flow Residual
Markov Detailed-Balance Flow Residual: Calculate detailed-balance residual from forward stationary transition flow and reverse stationary transition flow.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Markov Detailed-Balance Flow Residual to answer this question: calculate detailed-balance residual from forward stationary transition flow and reverse stationary transition flow? Enter forward stationary transition flow and reverse stationary transition flow; the calculator shows detailed-balance residual. For example: forward stationary transition flow=0.12 and reverse stationary transition flow=0.115 produce detailed-balance residual=0.0049999999999999906. The answer tells you detailed-balance residual.
Age 15Explain it to a 15-year-oldConnect it to the formula
Detailed balance requires paired stationary transition flows to match, so their difference is a reversibility residual. This page evaluates the relationship directly. The rule is c=a−b. Its input values are forward stationary transition flow, reverse stationary transition flow, and the main result is detailed-balance residual. For example: forward stationary transition flow=0.12 and reverse stationary transition flow=0.115 produce detailed-balance residual=0.0049999999999999906.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated markov detailed-balance flow residual relation over the valid real-number domain stated below. The implemented relation is c=a−b, evaluated from forward stationary transition flow, reverse stationary transition flow to produce detailed-balance residual. Detailed balance requires paired stationary transition flows to match, so their difference is a reversibility residual. This page evaluates the relationship directly. A nonzero residual rejects detailed balance for that pair but does not by itself measure total chain distance from reversibility.
Inputs and valid domain
- forward stationary transition flow must be a finite real number.
- reverse stationary transition flow must be a finite real number.
Important boundary: A nonzero residual rejects detailed balance for that pair but does not by itself measure total chain distance from reversibility.
The formula
c=a−b
How the calculator works through it
It substitutes forward stationary transition flow, reverse stationary transition flow into the formula and exposes every numerical step above. The main output is detailed-balance residual.
Read the result correctly
The detailed-balance residual is the direct answer to “calculate detailed-balance residual from forward stationary transition flow and reverse stationary transition flow.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
forward stationary transition flow=0.12 and reverse stationary transition flow=0.115 produce detailed-balance residual=0.0049999999999999906.
Where this model stops being reliable
A nonzero residual rejects detailed balance for that pair but does not by itself measure total chain distance from reversibility.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Markov Detailed-Balance Flow Residual works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Markov Detailed-Balance Flow Residual uses c=a−b. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Probability as a modelled proportion
Probability rules are needed to interpret what the Markov Detailed-Balance Flow Residual result says about possible outcomes.
Review this foundation about 5 min
Optional enrichment
- Ordered arrangements
Counting ordered arrangements can extend Markov Detailed-Balance Flow Residual to more detailed sample spaces and event models.
Review this foundation about 5 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read forward stationary transition flow, reverse stationary transition flow.
- Evaluate the principal relationship: c=a−b.
- Return detailed-balance residual and check the domain conditions described above.
Python
from math import *
def markov_detailed_balance_residual_calculator(a, b) -> float:
return (a - b)
assert abs(markov_detailed_balance_residual_calculator(0.12, 0.115) - 0.0049999999999999906) < 1e-6 * max(1.0, abs(0.0049999999999999906))
C
#include <assert.h>
#include <math.h>
double markov_detailed_balance_residual_calculator(double a, double b) {
return (a - b);
}
int main(void) {
const double expected = 0.0049999999999999906;
const double actual = markov_detailed_balance_residual_calculator(0.12, 0.115);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double markov_detailed_balance_residual_calculator(double a, double b) {
return (a - b);
}
int main() {
constexpr double expected = 0.0049999999999999906;
const double actual = markov_detailed_balance_residual_calculator(0.12, 0.115);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double markov_detailed_balance_residual_calculator(double a, double b)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global markov_detailed_balance_residual_calculator
section .text
markov_detailed_balance_residual_calculator:
push rbp
mov rbp, rsp
sub rsp, 32
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd xmm0, [rbp-8]
subsd xmm0, [rbp-16]
movsd [rbp-24], xmm0
movsd xmm0, [rbp-24]
leave
ret
MATLAB
function result = markov_detailed_balance_residual_calculator(a, b)
result = (a - b);
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_] := (a - b);
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Introductory Statistics 2e
Read the free OpenStax statistics textbookCite this book
- APA 7
- Illowsky, B., & Dean, S. (2023). Introductory statistics 2e. OpenStax. https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
- MLA 9
- Illowsky, Barbara, and Susan Dean. Introductory Statistics 2e. OpenStax, 2023, https://openstax.org/books/introductory-statistics-2e/pages/1-introduction.
- Chicago author-date
- Illowsky, Barbara, and Susan Dean. 2023. Introductory Statistics 2e. Houston, TX: OpenStax. https://openstax.org/books/introductory-statistics-2e/pages/1-introduction.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Markov Detailed-Balance Flow Residual Calculator. MW SysArc Tools. https://math.mwsysarc.com/probability/markov-detailed-balance-residual-calculator
MLA 9
MW SysArc. “Markov Detailed-Balance Flow Residual Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/probability/markov-detailed-balance-residual-calculator. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Markov Detailed-Balance Flow Residual Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/probability/markov-detailed-balance-residual-calculator.
Harvard
MW SysArc (2026) ‘Markov Detailed-Balance Flow Residual Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/probability/markov-detailed-balance-residual-calculator (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_markov_detailed_balance_residual_calculator_2026,
author = {{MW SysArc}},
title = {Markov Detailed-Balance Flow Residual Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/probability/markov-detailed-balance-residual-calculator},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Markov Detailed-Balance Flow Residual Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/probability/markov-detailed-balance-residual-calculator
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Markov Detailed-Balance Flow Residual do?
Calculate detailed-balance residual from forward stationary transition flow and reverse stationary transition flow.
How does the Markov Detailed-Balance Flow Residual work?
The calculator applies c=a−b. Detailed balance requires paired stationary transition flows to match, so their difference is a reversibility residual. This page evaluates the relationship directly.
What can I learn from the Markov Detailed-Balance Flow Residual?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .